An optimal linear controller design for an underactuated unicycle

In this work, we develop a unicycle that consists of a wheel and a pendulum. The control objective is for unicycle to track a given trajectory while keep the pendulum at the balanced position. In the unicycle, the only actuator is a motor mounted on the chassis, which generates torque to drive wheel. Hence the unicycle is an underactuated mechanism, and the control problem is challenging. The dynamic model of the unicycle is derived first and linearized around an equilibrium. Next, a LQR controller with full state feedback is designed based on the linearized model. Due to existence of parametric uncertainties and model mismatch such as the ignorance of actuator dynamics in the practical unicycle, LQR is unable to achieve desired control performance. Iterative learning tuning (ILT) method is introduced to adjust LQR so as to improve the control response iteratively. In this work, we propose an ILT law to tune the state weighting matrix in LQR objective function. The ILT law is derived by minimizing a scalar cost function with respect to the weighting matrix. Through simulation and experiment results, the effectiveness of the proposed LQR and ILT approach is validated.