Comprehensive Modeling and LQR Control of the Double Wheel on Wheel System

The inverted pendulum and other similar systems attract the attention of educators and researchers, serving as a benchmark for testing control strategies. Among these systems is the ball on wheel where the goal is to balance a ball on top of a driven wheel. We extend the ball on wheel concept and present for the first time the double wheel on wheel (DWOW) system, consisting of three wheels, one on top of the other, where the lower driven wheel tries to balance the other two. The purpose of this work is to describe, model and balance the DWOW system. The system, including the actuator and transmission, has been comprehensively modeled using three methods: Bond Graph, Lagrange and Simulink's Simscape; and some important features of each model have been discussed. The system has been balanced using an LQR controller, designed from a linearized model. The three models have been validated by comparing their simulations using normalized root mean squared errors (NRMSE), which took very small maximum values of \(6.44\times {10}^{-3}\) $6.44\times {10}^{-3}$ and \(0.138\times {10}^{-3}\) $0.138\times {10}^{-3}$ for the uncontrolled and controlled system, respectively. For the controlled system, the settling times were 3.66 and 3.61 s for the angles of the two upper wheels. The dynamic features of the DWOW system make it attractive for academia, both in education and research, although it could also find practical applications.