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To ensure the operational safety of nuclear power plants, we present a Quadruped Inspection Robot that can be used for many types of steam generators. Since the Inspection Robot relies on the Holding Modules to grip the tube-sheet, it can be regarded as a hybrid robot with variable configurations, switching between 4-RRR-RR, 3-RRR-RR, and two types of 2-RRR-RR, and the variable configurations bring a great challenge to dimensional synthesis. In this paper, the kinematic model of the Inspection Robot in multiple configurations is established, and the analytical solution is given. The workspace mapping is analyzed by the solution-space, and the workspace of multiple configurations is decomposed into the workspace of 2-RRR to reduce the analysis complexity, and the workspace calculation is simplified by using the envelope rings. The optimization problem of the manipulator is transformed into the calculation of the shortest contraction length of the swing leg. The switching performance of the Inspection Robot is evaluated by stride-length, turning-angle, and workspace overlap-ratio. The performance indexes are classified and transformed based on the proportions and variation trends of dimensional parameters to reduce the number of optimization objective functions, and Pareto optimal solutions are obtained using an intelligent optimization algorithm.

Flippers are key tissues through which frogs swim in water, and their spreading and contracting state directly affects their ability to move while driving their hind limbs. First, inspired by movement mechanism of frog flippers, a controllable soft-body extension-driven frog-like flipper mechanism was designed by combining the large elasticity and strong ductility of soft-body materials. This mechanism is driven by single motor via linkage gear, enabling it to simulate the changing state of flipper extension precisely controlled by the frog as it swims. Second, nonlinear mechanical calculations of flipper deformation membrane were carried out, and deformation optimization model based on Ogden's super soft-body material was established. Subsequently, integral analysis of the deformed area of flippers was used to obtain a fluid dynamics model of the paddling performance. Theoretical calculations and computational fluid dynamics simulations have shown that flippers can instantaneously control changes in the flipper area to inhibit fluid separation while swimming, effectively reducing the water drag of flippers in water and thus increasing the swimming propulsive force. Finally, when the spreading area of flippers was the same, and the swimming efficiency increased by 1, 1.6, and 1.9 times compared with that of fixed, passive folding, and passive spring flippers, respectively.

Crawling robots are the focus of intelligent inspection research, and the main feature of this type of robot is the flexibility of in-plane attitude adjustment. The crawling robot HIT_Spibot is a new type of steam generator heat transfer tube inspection robot with a unique mobility capability different from traditional quadrupedal robots. This paper introduces a hierarchical motion planning approach for HIT_Spibot, aiming to achieve efficient and agile maneuverability. The proposed method integrates three distinct planners to handle complex motion tasks: a nonlinear optimization-based base motion planner, a TOPSIS-based base orientation planner, and a Mask-D3QN (MD3QN) algorithm-based gait motion planner. Initially, the robot’s base and foot workspace were delineated through envelope analysis, followed by trajectory computation using Larangian methods. Subsequently, the TOPSIS algorithm was employed to establish an evaluation framework conducive to foundational turning planning. Finally, the MD3QN algorithm trained foot-points to facilitate robot movement along predefined paths. Experimental results demonstrated the method’s adaptability across diverse tube structures, showcasing robust performance even in environments with random obstacles. Compared to the D3QN algorithm, MD3QN achieved a 100% success rate, enhanced average overall scores by 6.27%, reduced average stride lengths by 39.04%, and attained a stability rate of 58.02%. These results not only validate the effectiveness and practicality of the method but also showcase the significant potential of HIT_Spibot in the field of industrial inspection.

The frog-inspired jumping robot is an interesting topic in the field of biomechanics and bionics. However, due to the frog’s explosive movement and large range of joint motion, it is very difficult to make their structure completely bionic. To obtain the optimal jumping motion model, the musculoskeletal structure, jumping movement mechanism, and characteristics of frogs are first systematically analyzed, and the corresponding structural and kinematic parameters are obtained. Based on biological characteristics, a model of the articular bone structure is created, which can fully describe the features of frog movement. According to the various factors affecting the frog’s jumping movement, mass and constraints are added, and the complex biological joint structure is simplified into four different jumping structure models. The jumping ground reaction force, velocity, and displacement of the center of mass, joint torque, and other motion information of these four models are obtained through ADAMS simulation to reveal the jumping movement mechanism and the influencing factors of frogs. Finally, various motion features are analyzed and compared to determine the optimal structural model of the comprehensive index, which provides a theoretical basis for the design of the frog-inspired jumping robot.

Inspired by the instantaneous power of the hindlimbs of frogs and the dexterous posture adjustment of their forelimbs, a frog-like jumping robot based on a soft body detonation drive was designed. By combining the flexibility of silicone material and the detonation characteristics of mixed hydrogen–oxygen gas, a soft body detonation driver was constructed to cause the robot to jump from the hindlimbs to achieve a high energy instantaneous explosive force. Moreover, a spring-based forelimb mechanism is designed. This mechanism not only adjusts the robot with different starting jump posture angles but also maintains a smooth landing by providing cushioning. The kinematics and efficiency models of jumping are also established by the D-H parameter method and geometric method, and they are verified and analysed. Finally, combined with the control system, an experimental prototype (weight 495 g) with a height of 135 mm, a trunk length of 120 mm, front and hindlimbs of 195 mm, and a width of 90 mm is developed. The experiment proved that the robot can not only achieve 6 consecutive jumps, a single jump height of 0.6 m and a spanning distance of 1.2 m but also easily cross rectangular obstacles.

Summary: According to the high-energy-density movement characteristics of animals during jumping, soft-body cavity-type detonation driver that combines the explosive chemical reaction mechanism of hydrogen and oxygen is designed, in order to control the robot in jump to achieve output optimization. Then, combined with the theoretical values of the detonation dynamic equation and experimental data for the performance parameters, the influences of the mixing ratio of hydrogen (H2) and oxygen (O2), the volume of mixed hydrogen and oxygen in the cavity, and the shape, wall thickness, and area ratio value of the soft-body cavity on the output performance of the detonation driver are analyzed. When gas volume is 20:10 mL, the jump height reaches 2.5 m. In addition, the upper and lower area ratio of cavity is optimized to 2:1, improving the output performance by 21.6% on average. Therefore, the above research results provide reference for the driver structure design of jumping robot.

Under the influence of nuclear radiation, the reliability of steam generators (SGs) is an important factor in the efficiency and safety of nuclear power plant (NPP) reactors. Motion planning that remotely manipulates an SG mobile tube-inspection robot to inspect SG heat transfer tubes is the mainstream trend of NPP robot development. To achieve motion planning, conditional traversal is usually used for base position optimization, and then the A∗ algorithm is used for path planning. However, the proposed approach requires considerable processing time and has a single expansion during path planning and plan paths with many turns, which decreases the working speed of the robot. Therefore, to reduce the calculation time and improve the efficiency of motion planning, modifications such as the matrix method, improved parent node, turning cost, and improved expanded node were proposed in this study. We also present a comprehensive evaluation index to evaluate the performance of the improved algorithm. We validated the efficiency of the proposed method by planning on a tube sheet with square-type tube arrays and experimenting with Model SG.

The motion process and force of the jumper crossing a multiphase environment are of great significance to the research of small amphibious robots. Here, CFD (Computational Fluid Dynamics)-based simulation analysis for motions through multiphase environments (water–air multiphase) is successfully realized by UDF (user-defined function). The analytical model is first established to investigate the jumping response of the jumpers with respect to the jump angle, force, and water depth. The numerical model of the jumper and its surrounding fluid domain is conducted to obtain various dynamic parameters in the jumping process, such as jumping height and speed. Satisfactory agreements are obtained by comparing the error of repeated simulation results (5%). Meanwhile, the influence of the jumper’s own attributes, including mass and structural size, on the jumping performance is analyzed. The flow field information, such as wall shear and velocity when the jumper approaches and breaks through the water surface, is finally extracted, which lays a foundation for the structural design and dynamic underwater analysis of the amphibious robot.

This paper focuses on the problem of extracting the physical dynamic parameters which are fundamental for computing the positive-definite link mass matrix. To solve this problem, a minimal set of dynamic parameters were firstly identified by the standard least squares method. In order to simplify the dynamics model, a new set of essential dynamic parameters were calculated by eliminating the poorly identified parameters with an iterative approach. Based on these dynamic parameters with better identification quality, a universally global optimization framework was proposed here to retrieve the set of physical dynamic parameters of a serial robot, in which parameter bounds, linear and nonlinear constraints with physical consistency can be easily considered, such as the triangle inequality of the link inertia tensors, the total link mass limitations, other user-defined constraints and so on. Finally, validation experiments were conducted on the KUKA LBR iiwa 14 R820 robot. The results show that the proposed optimization framework is effective, and the identified dynamic parameters can predict the robot joint torques accurately for the validation trajectories.

Frogs are vertebrate amphibians with both efficient swimming and jumping abilities due to their well-developed hind legs. They can jump over obstacles that are many or even tens of times their size on land. However, most of the current jumping mechanisms of biomimetic robotic frogs use simple four-bar linkage mechanisms, which has an unsatisfactory biomimetic effect on the appearance and movement characteristics of frogs. At the same time, multi-joint jumping robots with biomimetic characteristics are subject to high drive power requirements for jumping action. In this paper, a novel jumping mechanism of a biomimetic robotic frog is proposed. Firstly, the structural design of the forelimb and hindlimb of the frog is given, and the hindlimb of the robotic frog is optimized based on the design of a single-degree-of-freedom six-bar linkage. A simplified model is established to simulate the jumping motion. Secondly, a spring energy storage and trigger mechanism is designed, including incomplete gear, one-way bearing, torsion spring, and so on, to realize the complete jumping function of the robot, that is, elastic energy storage and regulation, elastic energy release, and rapid leg retraction. Thirdly, the experimental prototype of the biomimetic robotic frog is fabricated. Finally, the rationality and feasibility of the jumping mechanism are verified by a jumping experiment. This work provides a technical and theoretical basis for the design and development of a high-performance amphibious biomimetic robotic frog.

Soft modular robots have the advantages of both bionic continuum robots and modular self-reconfigurable robots. They have potential application for working in narrow space and uneven terrain. The outstanding abilities of the soft modular robots are infinite degrees of freedom and a changeable configuration. However, these advantages also incur difficulty in mathematical modeling and continuous motion control, especially for docking, which is the key to realizing a changeable configuration. Therefore, the development of a soft modular robot requires a modeling method to guide the module's motions accurately. In this paper, we design and manufacture a pneumatic soft modular robot with a novel connecting mechanism to achieve docking. A nonlinear dynamic model of our soft module is established. In particular, the analytic solutions of the module plane docking are presented. Based on the nonlinear dynamic modeling, three experiments are performed under different conditions. The experimental results pertaining to module plane bending agree with the theoretical values, which verify the accuracy of the modeling method. Moreover, fixed point docking and two modules docking experiments demonstrate the validity of the modeling method and the docking ability. The modeling method proposed in this paper can well guide the docking of soft modular robots.

A variable magnification ratio transmission structure powered by the electric actuators is proposed to improve the flexibility and portability of the exoskeleton under heavy load carrying condition. The parameters of connecting rod size and hanging position are optimized to ensure that the output torque of active joints can fully envelope the demand load area. The control strategy based on intrinsic sensing is designed to realize the automatic human motion intention prediction and flexible trajectory tracking. The newly developed split embedded connecting rod can accurately measure the human-robot interaction (HRI) force applied to the exoskeleton and extract the human motion intention without being affected by the differences in wearing status. The force tracking control based on the zero-force following is modified by feedforward compensation with extreme learning machine (ELM), which enhances the response speed to human motion intention and reduces the HRI force by 70.6%. Based on multi-sensor information, stacked autoencoder deep neural networks (DNNs) are utilized to realize the automatic locomotion transition and the corresponding control parameters' switching. After optimization by a hybrid algorithm of genetic algorithm and particle swarm optimization (GA_PSO), the identification accuracy is enhanced from 96.2% to 99.7%. The adaptive neural-fuzzy inference system (ANFIS) is used to analyze the plantar pressure to achieve flexible switching between the swing phase and the stance phase. The experiments under various gait motion trajectories assisted by novel weight-bearing exoskeleton are carried out for evaluation, and the performance of the proposed control strategy based on motion intention prediction, locomotion mode identification, and gait phase switching is effectively verified.

The vibration problem influences the precision of the equipment. However, some vibration isolation systems (VISs) with the perfect theoretical and simulation results are still unable to suppress the vibration well, resulting in the damage during the operation, such as the phenomenon of fragments for the end effector of a glass substrate handling robot. Therefore, this paper focuses on the glass substrate handling robot in actual production and simplifies the VIS into a 2D (2 degree) system. From the analysis of the deviation of material parameters and installation deviations between vibration isolators, this paper concludes that, as long as there is a deviation, the vibration isolation performance of the system will be reduced. Through the mutually verified theory, simulation, and experimental structure, this paper also presents that the resonance frequency increase factor of the vibration isolation system is approximately the deviation. As for the end effector, the distance of the two vibration isolations is 579.4 mm. The change of the load does not change the isolation frequency of the same system, but the resonance amplitude will increase. The results of this study can supervise the installation and selection of vibration isolators in VISs. At the same time, the reasons for the weak vibration isolation performance of some systems are explained.

The piezoelectric sensor-actuator plays an important role in micro high-precision dynamic systems such as medical robots and micro grippers. These mechanisms need high-precision position control, while the size of the sensor and actuator should be as small as possible. For this paper, we designed and manufactured a structure-integrated piezoelectric sensor-actuator and proposed its PID (Proportion Integral Differential) control system based on the dynamic hysteresis nonlinear model and the inverse model. Through simplifying the structure of the piezoelectric sensor-actuator by the centralized parameter method, this paper establishes its dynamic model and explores the input–output transfer function by taking the relationship between the output force and displacement as the medium. The experiment shows the maximum distance of the hysteresis curve is 0.26 μm. By parsing the hysteresis curve, this paper presents a dynamic hysteresis nonlinear model and its inverse model based on a 0.5 Hz quasi-static model and linear transfer function. Simulation results show that the accuracy of the static model is higher than that of the dynamic model when the frequency is 0.5 Hz, but the compensation accuracy of the dynamic model is obviously better than that of the static model with the increase of the frequency. This paper also proposes a control system for the sensor-actuator by means of the inverse model. The simulation results indicate that the output root mean square error was reduced to one-quarter of the original, which proves that the structure-integrated piezoelectric sensor-actuator and its control system have a great significance for signal sensing and output control of micro high-precision dynamic systems.

This paper presents the discrete state space mathematical model of the end-effector in industrial robots and designs the linear-quadratic-Gaussian controller, called LQG controller for short, to solve the low frequency vibration problem. Though simplifying the end-effector as the cantilever beam, this paper uses the subspace identification method to determine the output dynamic response data and establishes the state space model. Experimentally comparing the influences of different input excitation signals, Chirp sequences from 0 Hz to 100 Hz are used as the final estimation signal and the excitation signal. The LQG controller is designed and simulated to achieve the low frequency vibration suppression of the structure. The results show that the suppression system can effectively suppress the fundamental natural frequency and lower vibration of end-effector. The vibration suppression percentage is 95%, and the vibration amplitude is successfully reduced from ±20 μm to ±1 μm. The present work provides an effective method to suppress the low frequency vibration of the end-effector for industrial robots.

Jumping robots can be widely applied in archeology, interstellar probe, antiterrorism, and resource exploration. This article adopts the frog as the bionic object and briefly analyzes biological features and jumping movements in a complete jumping sequence of the frog. Then an equivalent six-bar mechanism model depicting frog jumping movement is built up, and the optimizing simulation of the six-bar mechanism model is performed. On the basis of the optimized results, the robot is designed and prototype is manufactured. An adaptive cascade controller is proposed to control the robot. Proportional–integral–derivative parameters of the controller are tuned online using radial basis function neural network to deal with nonlinearities of the robot. Jumping experiment demonstrates the jumping capacity of the robot and the ability of the proposed controller in tracking the desired trajectories.

The jumping robot has been a hot research field due to its prominent obstacle-climbing ability and excellent capacity in terrain adaptation and autonomous movement. However, huge impact between the robot and the ground when landing may cause structure damage, unbalanced movement, and even system crash. Therefore, trajectory planning of the jumping process has been a great challenge in robotic research, especially for the robot with varying underactuated and redundant joints. An intermittent jumping quadruped robot driven by pneumatic muscle actuators (PMAs) and owning variable redundant and underactuated joints designed in a previous study is taken as the study object. This paper divides the problem of trajectory planning into trajectory planning in the centroid space and joint space. Trajectory planning of different jumping phases in the centroid space adopts the scheme of minimizing the peak reaction force from the ground, then trajectory planning of the joint space is performed obeying the principle of minimizing consumed active torques. A jumping experiment is performed and validates the effectiveness of the proposed trajectory algorithm.

Estimate of the hydrodynamic force exerting on the frog is a key factor in design of mechanical structure and control system. To research propulsion generation mechanism during frog swimming, computational fluid dynamics–based method was performed. From the propulsive force results, drag and lift components of hydrodynamic forces were decomposed in terms of the palm motion. The thrust in the aquatic swimming trial turned to involve the lift propulsion, while drag component contributed almost the whole thrust in the terrestrial swimming trial. The hydrodynamic results were analyzed and compared to reveal the propulsion generation mechanism. The lift thrust characterized a U-shaped vortex ring around the palm to symbolize the lift mechanism.

Robot joint friction is an important and complicated issue in improving robot control performance. In this paper, a virtual sensor based on the total generalized momentum concept is proposed to estimate the total friction torque, including both the motor-side and link-side friction, of robot joints without joint torque sensors. The proposed algorithm only requires a robot joint dynamics model and not a complex friction model dependent on factors such as time and velocity. By compensating for the estimated friction torque with a robot joint controller, the trajectory tracking performance of the controller, especially the velocity tracking performance, can be improved. To verify the effectiveness of the developed algorithm, 2-DOF planar manipulator simulations and single-joint system experiments are conducted. The simulation and experimental results show that the designed virtual sensor can effectively estimate the total joint friction disturbance and that the controller trajectory tracking performance is improved after observed friction compensation. However, the position tracking performance improvement of the controller is less than that for the velocity tracking performance improvement during the experiments. In addition, the velocity step response ability and velocity tracking performance of the controller are improved more at low velocities than that at high velocities in the experiments. The proposed algorithm has engineering and theoretical significance for estimating robot joint friction and improving the performance of robot joint controllers.

Multi-clearance is the main cause for the performance and reliability decline of complicated mechanical systems. The increased clearance could induce contacts and impacts in joints, and consequently affect control accuracy. A nonlinear dynamic model of planetary gears with multi-clearance coupling is proposed in the current study to investigate the mechanism of influence of clearance on the dynamic performance. In addition, the coupling relationship between radial clearance and backlash is integrated into the multi-body system dynamics. The vibration characteristics of planetary gears with the changes of rotational velocity, clearance size and inertia load are explored. The numerical simulation results show that there are complex coupling relations in planetary gear systems, due to the multi-clearance coupling. The phenomenon of system resonance may occur with the changes of rotational velocities and clearances’ sizes. Multi-clearance coupling can significantly increase the resonant response of planetary gear systems in empty-load or light-load states.

Articulated caterpillar robot has various locomotion patterns—which make it adaptable to different tasks. Generally, the researchers have realized undulatory (transverse wave) and simple rolling locomotion. But many motion patterns are still unexplored. In this paper, peristaltic locomotion and various additional rolling patterns are achieved by employing sinusoidal oscillator with fixed phase difference as the joint controller. The usefulness of the proposed method is verified using simulation and experiment. The design parameters for different locomotion patterns have been calculated that they can be replicated in similar robots immediately.

An enhanced model is proposed to describe static property of commercial braided pneumatic muscle actuators by including several important influencing factors. Elasticity of elastomer tube is considered and Ogden strain energy function is employed to describe its strain energy density. During pressurized process, small deformation of fiber occurs and is calculated using force balancing principle. Frictional forces within muscles are studied, which consist of friction within braid and that between bladder and braid. Isobaric experiments are performed and results verify the validity of the model.

For understanding the mechanism of frog swimming under water and designing a frog-inspired swimming robot, kinematics of the frog body and trajectories of joints should be obtained. In this paper, an aquatic frog, Xenopus laevis , was chosen for analysis of swimming motions which were recorded by a high speed camera, and kinematic data were processed in a swimming data extraction platform. According to the shape features of the frog, we propose a method that the frog eyes are set as the natural data extraction markers for body motion, and kinematic data of joint trajectories are calculated by the contour points on the limbs. For the data processing, a pinhole camera model was built to transform the pixel coordinate system to world coordinate system, and the errors caused by the water refraction were analyzed and corrected. Finally, from the developed data extraction platform, the kinematic data for the analysis of swimming mechanism and design of frog-inspired robot were obtained.

Braided pneumatic muscle actuator shows highly nonlinear properties between displacements and forces, which are caused by nonlinearity of pneumatic system and nonlinearity of its geometric construction. In this paper, a new model based on Bouc-Wen differential equation is proposed to describe the hysteretic behavior caused by its structure. The hysteretic loop between contractile force and displacement is dissolved into linear component and hysteretic component. Relationship between pressure within muscle actuator and parameters of the proposed model is discussed. A single degree of freedom manipulator actuated by PMA is designed. On the basis of the proposed model, a novel cascade position controller is designed. Single neuron adaptive PID algorithm is adopted to cope with the nonlinearity and model uncertainties of the manipulator. The outer loop of the controller is to handle position tracking problem and the inner loop is to control pressure. The controller is applied to the manipulator and experiments are conducted. Results demonstrate the effectiveness of the proposed controller.

A distributed control algorithm based on a stimulus pulse signal is proposed in this paper for the locomotion of a Modular Self-reconfigurable Robot (MSRR). This approach can adapt effectively to the dynamic changes in the MSRR's topological configuration: the functional role of the configuration can be recognized through local topology detection, dynamic ID address allocation and local topology matching, such that the features of the entire configuration can be identified and thereby the corresponding stimulus signals can be chosen to control the whole system for coordinated locomotion. This approach has advantages over centralized control in terms of flexibility and robustness, and communication efficiency is not limited by the module number, which can realize coordinated locomotion control conveniently (especially for configurations made up of massive modules and characterized by a chain style or a quadruped style).

Background: Although Pneumatic Artificial Muscle (PAM) has a promising future in rehabilitation robots, it's difficult to realize accurate position control due to its highly nonlinear properties.Objective: This paper deals with position control of PAM.Methods: To describe the hysteresis inside PAM, a polynomial based phenomenological function is developed. Based on the phenomenological model for PAM and analysis of pressure dynamics within PAM, an adaptive cascade controller is proposed. Both outer loop and inner loop employ BP Neural Network tuned PID algorithm. The outer loop is to handle high nonlinearities and unmodeled dynamics of PAM, while the inner loop is responsible for nonlinearities caused by pressure dynamics.Results: Experimental results show high tracking accuracy as compared with a convention PID controller.Conclusion: The proposed controller is effective in improving performance of PAM and will be implemented in a rehabilitation robot. [ABSTRACT FROM AUTHOR]

Background: When using upper limb exoskeletons that assist the movement of physically weak people, safety should be the most important index.Objective: In this paper, a passively safe, cable-driven upper limb exoskeleton with parallel actuated joints, which perfectly mimics human motions, is proposed.Methods: Compared with the existing upper limb exoskeletons which are mostly designed only considering the realization of functional properties, and having poor wearabity, a passively safe prototype for motion assistance based on human anatomy structure has been developed in our design. This design is based on the prior exoskeleton structure with the adoption of a gravity balanced device.Results and Conclusion: The gravity balanced mechanism was confirmed in theory and simulation, showing it has a positive effect on balance. [ABSTRACT FROM AUTHOR]

Muscular rigidity and atrophy caused by long-term underactivity usually lead to foot drop, strephenopodia, foot extorsion or some other complications for the lower limb movement disorders or lower limb surgery sufferers. The ankle-foot orthosis can help patients conduct the right ankle motion mode training, inhibit spasm and prevent ankle complications. In this paper, a quasi-passive 3 DOFs ankle-foot wearable orthosis was designed on the basis of kinematics and dynamics analysis of the ankle joint. Ankle joint trajectory and dynamic characteristics similar to those of natural gait can be obtained by the combination of passive energy storage and additional power complement. In terms of function, the orthosis has shock absorption and low energy consumption. Given its excellent characteristics of comfortableness, lightweight, and anthropomorphic construction, the orthosis can be used in medical institutions for rehabilitation training or as a daily-walking auxiliary equipment for surgery sufferers. [ABSTRACT FROM AUTHOR]

Aquatic and terrestrial animals have different swimming performances and mechanical efficiencies based on their different swimming methods. To explore propulsion in swimming frogs, this study calculated mechanical efficiencies based on data describing aquatic and terrestrial webbed-foot shapes and swimming patterns. First, a simplified frog model and dynamic equation were established, and hydrodynamic forces on the foot were computed according to computational fluid dynamic calculations. Then, a two-link mechanism was used to stand in for the diverse and complicated hind legs found in different frog species, in order to simplify the input work calculation. Joint torques were derived based on the virtual work principle to compute the efficiency of foot propulsion. Finally, two feet and swimming patterns were combined to compute propulsive efficiency. The aquatic frog demonstrated a propulsive efficiency (43.11%) between those of drag-based and lift-based propulsions, while the terrestrial frog efficiency (29.58%) fell within the range of drag-based propulsion. The results illustrate the main factor of swimming patterns for swimming performance and efficiency., (© 2017. Published by The Company of Biologists Ltd.)