Power Generation Performance of a Spherical Robot With Pendulums in Amphibious Environment

SUMMARY The energy of mobile robots severely limits their range of motion and work capabilities. This paper proposes a method of capturing energy from the amphibious environment for a spherical robot with pendulums. The movement of pendulums is analyzed during amphibious movement, and a feasible scheme is proposed for a pendulum to capture environmental energy and convert mechanical energy into electrical energy. The mathematical model of the swing power generation is established based on the pendulum dynamic equation and voltage balance equation. The physics experiment platform and virtual experimental platform are built to analyse the power generation performance. Furthermore, the power generation mathematical models are established respectively for the spherical robot rolling on the slope and floating in the water, and the power generation performance is analyzed and summarized under different conditions. The results show that the proposed power generation method and scheme can effectively supply the energy to the spherical robot, can enhance the endurance of the movement in the amphibious environment, and provide theoretical guidance for the development of the physical prototype of the new generation of amphibious spherical robot.

and the three-dimensional model is shown in Fig.(b). The outside of the robot is sealed by a transparent spherical shell, and all the electronic components are placed inside the spherical shell. Through the center of the spherical shell, a propeller propulsion device is installed inside the spherical shell. The left and right sides of the inner tube of the spherical shell are equipped with heavy pendulums, which can swing in two directions under the drive of the motors. In addition, a flywheel device is installed inside the spherical shell , Which is directly driven by a brushless motor. In short, the robot has three driving devices, and the structural design ensures that the rotation axis of the propeller, the pendulum and the flywheel are perpendicular to each other. As shown in Fig.1, the spherical robot has the ability to move in amphibious environments. On land, it relies on the rolling mode of the spherical shell to move. The rolling torque of the spherical shell is provided by the drive motor of heavy pendulums to realize the straight travel and turning of the robot. When the robot moves in the water, the propeller inside the duct provides thrust. The heavy pendulums device inside the spherical shell can generate pitching torque, the flywheel device can generate steering torque, and the three driving devices cooperate to control the direction of the robot's underwater movement to achieve flexible movement of 6 degrees of freedom underwater three-dimensional space. The spherical robot has the outstanding motion capabilities and is suitable for performing observation tasks in an amphibious environment. The spherical shell of the robot is closed to protect the internal electronic components, and the spherical shell has strong pressure resistance. The amphibious athletic ability allows it to adapt to more complex environments without overturning problems. Although the structure of the spherical robot can move amphibiously, it has a strong load capacity, and can carry a lot of sensing equipment, but it also has shortcomings. The movement resistance is relatively large. The heavy pendulum device is enclosed in the spherical shell. Frequent swings are required to adjust the pitch angle, which occupies a large internal space, so that the spherical shell cannot be fully loaded with batteries. Therefore, it is of great significance to improving the endurance based on the structure and motion characteristics of the spherical robot to capture environmental energy and utilization methods.
The pendulum device in the internal structure of the spherical robot plays an important role in the amphibious motion of the spherical robot. The state of rolling on the land is shown in Fig 2. Under the action of the motor torque M, the heavy pendulum swings forward by an angle α around the axis of the tube inside the spherical shell, and the reaction torque of the motor drives the spherical shell to roll forward at an angular velocity ω. The spherical shell parameters are set as follows: the center velocity v, the radius is R, the friction force is f, the mass is m 1 , the moment of inertia is J, and the horizontal perturbation force F y of the spherical shell caused by the heavy pendulum. Without sliding, the rolling equation of the robot in the y-axis direction can be expressed in formula (1).
Land Heavy pendulum f The state of the spherical robot in the water is shown in Fig.3. Under the action of the motor servo torque M, the heavy pendulum inside the spherical shell swings forward by an angle α around the axis in the longitudinal section. The reaction force of the motor servo controls the pitch angle of the spherical shell. The pipe axis is located in the longitudinal section. The propeller generates thrust T through the center of the sphere. The robot overcomes the water resistance F D and sails forward at speed v. The pendulum has horizontal disturbance force F x to the spherical shell. The water resistance coefficient of translation is C d , and the additional mass coefficient is λ. The rotating hydrodynamic coefficient of the spherical shell is very small and negligible. The x-axis motion equation of the spherical robot in water is shown in formula (2).
Based on the above model and principle analysis of the spherical robot rolling on land and moving in water, it can be seen that the driving torque M of the pendulum device plays an important role, and the reaction torque can be used to control the spherical shell rolling or attitude adjustment. The mass of the pendulum block is m 2 , and the length of the pendulum rod is l. The dynamic equation of the pendulum is shown as follows .
The amphibious environment contains abundant energy, such as terrain energy and wave energy. Based on the unique heavy pendulum structure of the spherical robot and the characteristics of the swing motion inside the spherical shell, a method for capturing and using environmental energy can be proposed. When a spherical robot rolls on land in the task of observing the slope environment, the relative rotation of the heavy pendulum and the spherical shell can be used to generate electricity. When the spherical robot performs the observation task in the water with waves, the relative swing of the heavy pendulum and the spherical shell can be used to generate electricity. The capturing environmental energy and utilizing method requires that the environmental energy be converted into the mechanical energy of the heavy pendulum, and then the generator is used to convert the mechanical energy into electrical energy. The electrical energy can be used by the sensor or stored in the battery, so The scheme of the energy capture device is mainly composed of two parts, which includes a suspension pendulum and a generator. Adopting this scheme has great advantages. The spherical robot uses the original heavy pendulum to capture terrain energy and wave energy, and does not need to design an additional power generation structure, which is easier to implement and does not affect the original motion performance of the robot.

Power Generation Model and Comparative Experiment
During the analysis of the principle of power generation, the spherical shell of the robot is set to be stationary, and the performance of the pendulum mechanical energy converted into electrical energy is analyzed. First, the theoretical model of the pendulum power generation is established. The power generation device is equipped with a heavy pendulum, pendulum shaft, speed increaser and generators. The pendulum shaft is restricted by the spherical shell hinge without translational motion.
The pendulum can rotate around the pendulum shaft. One end of the pendulum shaft is connected to a speed increaser, and the output end of the speed increaser is connected to input shaft of the generator, which can directly drive the generator to generate power, and the speed increaser is used to match the rotational speed of the pendulum shaft and the generator shaft. The power generating device can be reused as the driving device when the robot is moving. In this situation the generator is used as the motor, and the speed increaser is used as the reducer, so this design will not affect the amphibious movement ability of the robot. In the mechanism as shown in Fig.4, the gear ratio of the speed increaser is i. The rotational speed of the generator input shaft is ω2. The rotational speed of the pendulum shaft is ω1. The swing damping coefficient is c. The swing resistance torque is M. When the pendulum structure captures environmental energy into mechanical energy, it will produce variable-speed swing, and its motion equation is shown in Equation 4.
The energy harvesting device converts the mechanical energy of the pendulum into electric energy during the power generation process, and the electromagnetic torque T e of the generator is transmitted to the pendulum shaft through the speed increaser, which hinders the swing of the pendulum. According to the principle of electromagnetic induction, it can be known that the induced electromotive force is related to the rate of change of the magnetic flux. The motor winding is composed of N coils, and the angle between the coil and the magnetic field line is φ, so the induced electromotive force E at both ends of the motor winding can be further obtained by accumulation as follow: The generator of the robot uses a DC motor, so the magnetic field B and the coil area S are fixed values, K e is the electric potential coefficient of the motor, and ω 2 is the rotation speed of the generator input shaft, then the induced electromotive force can be further simplified into the following form: The resistance of the motor winding is R1, the current of the winding is ia, and the load circuit resistance is R2. According to the voltage balance equation of the motor, the relationship between the induced electromotive force E and the output voltage U can be obtained: The torque constant of the motor is K t . According to the principle of the generator, the expression of the electromagnetic torque T e can be obtained as follows: e t a T K i = (8) When the generator parameters have been determined, the combined formulas (6)~(8) can conclude that the electromagnetic torque T e is a function of the generator input shaft speed ω 2 . The electromagnetic torque is transmitted to the swing shaft and becomes larger through the speed increaser. The magnification is the speed increase ratio i=ω 2 /ω 1 , so it can be concluded that the damping torque T c is a function of the speed ω 1 at the input end of the speed increaser.
The equivalent damping coefficient is c 2 , and the electromagnetic damping torque T c can be expressed by the following formula: Considering that swing is also affected by mechanical damping, the mechanical damping coefficient is c 1 , then: In order to determine the damping parameters and power generation characteristics of the pendulum, a power generation experimental platform was built as shown in Fig.5. The platform has two parts: a motion mechanism and an energy harvesting device. The motion mechanism is mainly composed of horizontal linear guides, a motor and a sliding bearing seat, and the energy capture device is mainly composed of a pendulum block, a pendulum shaft and a generator. The measuring device mainly includes a gyroscope for measuring the swing angle and an oscilloscope for measuring the output voltage of the generator. First, R=0Ω and R=100Ω resistors with different resistance values were connected to the generator output terminal. The motor of the platform was braked to prevent the sliding bearing seat from moving, and the initial value of the swing angle α=0.5π, then let the heavy pendulum swing freely under the action of gravity and damping torque. The gyroscope measures the swing angular velocity ω 1 , and the oscilloscope measures the generator output voltage U. It is known that the generator and the pendulum shaft are connected by a speed increaser with i=40:1. The mass of the pendulum block is m 2 =2.7 kg, and the pendulum length l=0.2m. The experimental test data results are shown in the curve in Fig.6. It can be seen from Fig.6(a) that the swing speed gradually decreases under the action of damping. When R=0, the mechanical damping force is applied, and the swing energy decays slowly. When R=100Ω, the mechanical damping and electromagnetic damping work together to attenuate it. The mechanical energy of the pendulum is completely exhausted by the damping force within 12s. According to the functional principle, the following equation can be obtained: The mechanical damping coefficient and the electromagnetic damping coefficient are expressed: The known parameters and the test data in Fig.6 are brought into the above formula. The unknown parameters in the power generation model can be calculated. In order to facilitate the subsequent analysis of the power generation characteristics of the amphibious motion of the spherical robot, a virtual swing power generation prototype is built as shown in Fig.7. The virtual prototype is set with all parameters, and the simulation and experimental data are compared. The consistency of the results shows that the virtual prototype simulation is effective to analyze power generation characteristics. Change the load resistance R=125Ω at the output end of the generator. The other conditions are the same as the previous experiment. The experimental result is shown in Fig.8. The experiment is carried out again by controlling the motor of the platform, which drives the sliding bearing seat and the pendulum shaft to reciprocate on the slide rail, and the heavy pendulum continuously swings around the pendulum shaft under the action of inertial force and gravity. The experimental measurement results are shown in Fig.9.It can be seen that when the resistance R is set to 125 Ω and the pendulum shaft does not move in translation, the amplitude of the swing is gradually attenuated under the action of mechanical damping and electromagnetic damping. When the platform motor applies continuous reciprocating motion, the swing of the pendulum is also continuously output, which will not attenuate in the end. The comparison experiment and simulation results are basically the same, which shows that the virtual prototype simulation of the power generation model can effectively replace the physical experimental platform for subsequent power generation characteristics analysis.

Power Generation Performance of Robot Rolling on a Slope
The swing power generation module is integrated into the spherical robot body for rolling power generation. When the spherical robot rolls down on a slope, the gravitational potential energy is converted into kinetic energy, and the swing power generation module converts the kinetic energy of the spherical shell into electrical energy. It can stabilize the rolling speed. The rolling and power generation performance of the spherical robot is analyzed. First, a mathematical model is established to determine the influencing parameters. The schematic diagrams of the force on the spherical shell and the heavy pendulum are shown in Fig.10. The force is shown in Fig.10(a). The spherical robot with radius R rolls forward on the slope surface of θ angle at an angular velocity v. At the contact point between the spherical shell and the slope surface, the slope surface has the friction force f and the supporting force N on the spherical shell. The internal pendulum mechanism of the robot system will also apply component force F x , component force F y and damping moment T c to the robot, and the spherical shell part of the robot itself is also affected by gravity G. The moment of inertia of the spherical shell is J. According to Newton's Euler principle, the spherical shell mechanics equation can be obtained as follow when the spherical robot rolls on a slope to generate electricity : In addition to the spherical shell part, the force analysis of the internal pendulum mechanism should be performed for the force analysis of the spherical robot system. In the same way, the force analysis with the pendulum as the research object is shown in Fig.10(b). The pendulum structure of length l swings backward by an angle of α. When the heavy pendulum rotates along the central axis of the robot, it will be affected by the spherical shell reaction component x F , force component y F and damping moment c T  at the connection point, and the pendulum itself will also be affected by the gravity m2g. Ignoring the influence of air resistance and the quality of the pendulum rod, the force equation of the internal pendulum mechanism can be obtained according to Newton's Euler theorem when the spherical robot is rolling on a slope to generate electricity as follows: In formulas (15) and (16) It can be seen from the above formula (15), (16) and (17),that the damping torque T c will affect the rotation of the shell and the swing of the pendulum. The robot movement is controlled by changing the damping torque parameters to obtain a constant speed and power generation energy. The spherical shell and slope environment are added to the previous power generation simulation model for further simulation analysis.  In order to analyze the influence of the slope, the rotation speed and the power generation performance are obtained with the mass of the shell m 1 =15 kg, the mass of the pendulum m 2 =15 kg, and the damping coefficient c 2 =1 by simulation as shown in Fig.11. Formula (18) is used to calculate the power generation. ω s and ω p are the rotation speed of the shell and pendulum. From Fig.11(a), (c) and (e), it can be seen that the robot rolls down under the action of gravity force on the slope, and the speed of the shell gradually stabilizes, The stable speed becomes increases from 5 rad/s to 10 rad/s with the increase of the slope. But the speed of the pendulum tends to 0. From Fig.11(b), (d) and (f), it can be seen that the robot continues to generate power when it moves on the slope, and the power generation gradually stabilizes. The power generation increases with the increase of the slope. When the slope is 10/180 π rad, the power reaches more than 100 W.
In order to analyze the influence of the damping coefficient, the rotation speed and the power generation performance are obtained with the slope θ=10/180π rad by simulation as shown in Fig.12. From Fig.12(a), (c) and (e), it can be seen that the steady speed becomes decreases with the increase of the damping coefficient. From Fig.12(b), (d) and (f), it can be seen that the power generation decreases with the increase of the damping coefficient. When the damping coefficient c2=2, the speed of the shell starts to maintain a stable 5 rad/s after 2s, and the power generation can still maintain more than 50 W.

5.Power Generation Performance of Robot Floating in the Water
The spherical robot studied in this paper has the ability of amphibious motion. Therefore, based on the above research on the power generation characteristics of rolling on the slope, the characteristics of transforming wave energy in the water to generate electricity are further studied. When the spherical robot performs tasks in the water, the spherical shell Impacted by the wave force, the internal heavy pendulum will swing, which will drive the generator to generate electricity. The force of the robot moving in the water is shown in Fig.13. In order to establish the underwater power generation model of the spherical robot, the wave motion form must be described mathematically. The complex wave motion in the ocean is difficult to accurately express with mathematical equations. For the convenience of research and analysis, the wave motion form can be processed based on the linear wave theory, and the small amplitude waves will undulate in a simple harmonic form. Hw is the amplitude of the wave, Lw is the length of the wave, and fw is the frequency of the wave. Therefore, the horizontal velocity Uwx and the vertical velocity Uwz of the wave can be expressed as the following equation: The motion of the underwater robot is closely related to the wave hydrodynamic force acting on the spherical shell. For this small-scale structure, the Morision equation is mainly used to calculate the wave force. Therefore, this small spherical robot ignores the rotating water of the spherical shell. The horizontal hydrodynamic force Fwx and the vertical hydrodynamic force Fwz can be expressed as the following equation: In equation (20), ρ is the density of the fluid, C d is the drag coefficient of the spherical shell, A is the cross-sectional area of the robot, M a is the fluid additional mass coefficient of the spherical shell, U x is the horizontal movement speed of the robot, and U z is the vertical movement speed of the robot. The spherical shell is used as the research object as shown in Fig.13(a). The buoyancy of water is B. The mechanical equation is established as follows: When the robot is excited by waves to make a variable speed motion, the internal pendulum will swing as shown in Fig.13(b) with the rod length l and the swing angle α. At the connection point of the rotating shaft in the center of the spherical shell, the pendulum will be exerted by the spherical shell with the force x Fand y F . Ignoring the influence of air resistance and rod mass, the equivalent mass of the pendulum will be affected by the force of gravity m2g. According to Newton's law, we can get the force expression of the internal pendulum when the spherical robot moves in the water as shown in follow equation: In equations (21) and (22), the force Fx and x F , the force Fy and y F , the damping torque Tc and c T are the acting forces and the reaction forces that appear in pairs. According to Newton's third law, the magnitude of these three pairs of forces and moments is respectively equal. The mechanics equation (22) of the pendulum is brought into the mechanics equation (21)  It can be seen in the equation (21), (22) and (23), that the damping torque T c will affect the motion of the spherical robot in water. The rotation of the shell and the swing of the pendulum will also be affected. According to formula (9), the damping torque of the robot can be adjusted to obtain different movement speeds and power generation energy. Furthermore, the spherical shell and water environment are added to the previous simulation model for power generation analysis. In the time period between 10s and 20s , the motion and power generation data are shown in the Fig.14, Fig.15 and Fig.16.
The movement of the spherical robot in the water is affected by the amplitude and frequency of the waves, and the damping force will also have an effect. In order to analyze the influence of amplitude on power generation characteristics, a simulation was carried out with the frequency f w =0.6HZ and the damping coefficient c 2 =1. From Fig.14 (a), (c) and (e), it can be seen that the rotation speed of the robot's shell and pendulum are fluctuating, and the fluctuation frequency is consistent with the wave frequency. When the wave amplitude increases from 0.2m to 0.4m, the robot's rotational speed fluctuates sharply. The swing amplitude of the pendulum is greater than that of the shell, and the overall mechanical energy of the robot increases. From Fig.14 (b), (d) and (f), it can be seen that the generated power cannot remain stable, but the statistical average power is gradually increasing. When the wave amplitude is 0.2m, the generated power is 1.7W. When the wave amplitude increased to 0.6m, the power generation reached about 13W. In order to analyze the influence of frequency on power generation characteristics, a simulation was carried out with the amplitude H w =0.4m and the damping coefficient c 2 =1. From Fig.15 (a), (c) and (e), it can be seen that the robot's rotational speed fluctuates slowly when the wave frequency decreases from 0.5HZ to 0.3HZ, and the overall mechanical energy of the robot decreases. From Fig.15 (b), (d) and (f), it can be seen that the statistical average power is gradually decreasing. When the wave frequency is 0.5HZ, the generated power is 2W. When the wave frequency decreased to 0.3HZ, the power generation is about 0.02W.
In order to analyze the influence of damping coefficient on power generation characteristics, a simulation was carried out with the amplitude H w =0.4m and the frequency f w =0.6HZ. From Fig.16 (a), (c) and (e), it can be seen that the robot's rotational speed fluctuates sharply when the damping coefficient decreases from 0.8 to 0.4, and the overall mechanical energy of the robot increases. From Fig.15 (b), (d) and (f), it can be seen that the statistical average power is gradually increasing. When the damping coefficient is 0.8, the generated power is about 13W. When the damping coefficient decreased to 0.4, the power generation is about 17W.

Conclusion
In order to enhance the endurance of mobile robots, the developed amphibious spherical robot has good environmental adaptability. A feasible power generation scheme is proposed based on the driving structure. The motion and power generation characteristics are analyzed in the amphibious environment, and the conclusions mainly include the following point.
(1) By analyzing the structure of the spherical robot and the principle of motion in the amphibious environment, an energy supply scheme is proposed, which consists of a heavy pendulum and a generator. The terrain energy and wave energy are firstly converted into the mechanical energy of the pendulum, and then the generator converts mechanical energy into electrical energy. (2) A pendulum power generation model is built, and a physical experiment platform was built to verify the feasibility of the power generation scheme. The related parameters are solved. Further, a virtual platform is built to compare the experimental data and verify the effectiveness of the simulation model. (3) The dynamic model of the spherical robot is built on a slope, and the power generation performance is analyzed by simulation. It is concluded that the stable speed and the generation power of the robot increase when the slope angle increases. the generation power can reaches to 120W, and increasing the damping coefficient will have the opposite result. (4) The dynamic model of the spherical robot is built in the water. The fluctuation speed of the robot is shown by simulation. The fluctuation amplitude and the generation power become larger when the amplitude of the wave increases. Reducing the wave frequency will get the opposite result, but reducing the damping will increase the power to about 17W. (5) From the obtained formulas and results, it can be seen that adjusting the damping of the robot will have an impact on the robot's motion speed and power generation. However, the control of damping is closely related to the motor parameters and the transmission ratio.It is to focus on research to design these parameters for the next generation of robot.