Inverted pendulums are traditional dynamic problems. If an inverted pendulum is used in a moving cart, a new type of exciting issues will appear. One of these problems is two-wheeled inverted pendulum systems. Because of their small size, high performance in quick driving, and their stability with controller, researchers and engineers are interested in them. In this paper, a new configuration of one specific robot is modeled, and its dynamic behavior is analyzed. The proposed model can move in two directions, and with a proper controller can keep its stability during the operation.  In this robot, two cantilever beams are mounted on the two-wheeled base, and they can freely rotate around it. These two flexible cantilever beams act as flexible inverted pendulums.

The mathematical model represents the response of the system to small disturbances from piezoelectric actuators that are attached to the base of each pendulum. These actuators cause deformation and bending in the continuous pendulums when voltage is applied to them. The mathematical model of this system is obtained using the extended Hamilton's Principle. The results show that the governing equations of motion are highly nonlinear and contain several coupled partial differential equations (PDEs). In order to extract the natural modes of the beams, the undamped, unforced equations of motion and boundary conditions of the beams are used. If a limited number of modes (*N~1~* and *N~2~*) are selected for each beam, the coupled PDEs will be changed to *N~1~+N~2~+5* ordinary differential equations (ODEs).

To investigate the dynamic behavior of the system, the equations of motions are solved numerically, and the results are presented in different scenarios. First, a sweep frequency input is applied to the system to extract the natural frequencies of the system. This standard input provides a fairly uniform spectral excitation and gives the modes of the systems. The result shows the spectral analysis of the time history of the system and its natural frequencies. In the next scenario, the lateral movement of the system in *X*-direction is presented by applying two same unit step input voltages to the piezoelectric actuators. The final scenario is the rotation test in the *X-Y* plane. In this test, two-step input voltages with different amplitudes are applied to the piezoelectric actuators. This makes different beams' lateral vibrations and bases' rotational angles, and consequently, different torques applied to the wheels. This causes different angular velocities in the wheels and produces a rotational angle in the robot system. The simulation shows that the governing equations are correct, and the system is ready for designing a proper controller. It should be mentioned that in the future works, the derived equations will be validated experimentally, and a suitable control strategy will be applied to the system to make it fully automated and more applicable.


A New Dynamic Model of a Two-Wheeled Two-Flexible-Beam Inverted Pendulum Robot

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

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