In this paper we develop the control and stabilization of a spherical jointed inverted pendulum balanced on an omnidirectional cart. The system consists of an omnidirectional cart with mecanum wheels equipped with a spherical inverted pendulum attached at the center of the platform. The inverted pendulum is free to fall in any direction perpendicular to the horizontal plane. The omnidirectional cart has the special ability to move in any direction without changing orientation. It can also rotate around its vertical axis. This balancing platform provides a base with holonomic motion to support and balance the pendulum. In this work, the system has been decoupled into two separate subsystems in the x-z and y-z plane. We develop the system dynamic equations in both vertical planes and design a LQR controller to stabilize the system. Using one-dimensional pendulum experimental data, we validate our controller and extend the approach to stabilize the spherical pendulum in both vertical directions. Simulation results are presented to show the effectiveness of the decoupled system LQR controller in stabilizing the spherical pendulum.
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