<abstract xmlns="http://www.w3.org/1999/xhtml"><p> In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2- DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.</p></abstract>

In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated gyroscope. First, the equations of motion of the complete system (3-DOF) are presented in both Lagrangian and Hamiltonian formalisms. Moreover, the conditions to reduce the system from a 3-DOF to a 2- DOF gyroscope, by using Routh’s equations of motion, are shown. Next, the solutions of the partial differential equations involved in getting the proper controller are presented using a reduction method to handle them as ordinary differential equations. Besides, since the gyroscope has no potential energy, it presents the inconvenience that neither the desired potential energy function nor the desired Hamiltonian function has an isolated minimum, both being only positive semidefinite functions; however, by focusing on an open-loop nonholonomic constraint, it is possible to get the Hamiltonian of the closed-loop system as a positive definite function. Then, the Lyapunov direct method is used, in order to assure stability. Finally, by invoking LaSalle’s theorem, we arrive at the asymptotic stability of the desired equilibrium point. Experiments with an underactuated gyroscopic mechanical system show the effectiveness of the proposed scheme.

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

In this paper we present interconnection and damping assignment passivity-based control (IDA-PBC) applied to a 2 degrees of freedom (DOFs) underactuated...

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

Published Online: Jan 11, 2019

Page range: 661 - 677

Received: Nov 23, 2017

Accepted: Jul 15, 2018

DOI: https://doi.org/10.2478/amcs-2018-0051

Keywords
gyroscope device, gyroscopic forces, cyclic coordinates, generalized momenta, Routh’s equations, IDA-PBC

© 2019 Gustavo Cordero, et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Interconnection and Damping Assignment Passivity–Based Control of an Underactuated 2–DOF Gyroscope

Published Online: Jan 11, 2019

Page range: 661 - 677

Received: Nov 23, 2017

Accepted: Jul 15, 2018

DOI: https://doi.org/10.2478/amcs-2018-0051

Keywordsgyroscope device, gyroscopic forces, cyclic coordinates, generalized momenta, Routh’s equations, IDA-PBC

© 2019 Gustavo Cordero, et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
gyroscope device, gyroscopic forces, cyclic coordinates, generalized momenta, Routh’s equations, IDA-PBC