Periodic trajectory generation and tracking control for a class of underactuated mechanical systems

The underactuated mechanical systems (UMS) is a special class of nonlinear systems with fewer inputs than its degree-of-freedoms (DOF). Currently the research work of such systems mainly focuses on the equilibrium point stabilization and periodic trajectory tracking problem. The problem of steering a UMS to arbitrary points in its state space is hardly studied. Although the existing periodic trajectory generation research work has demonstrated the possibility of generating periodic orbits that can be utilized to realize point-to-point steering control, no systematic methodology of obtaining a trajectory to pass through a specified point is proposed. Furthermore the existing LQR based trajectory tracking controller that requires analytical solutions for the periodic orbit function and nonlinear Ricatti equation is hard to implement. Therefore, in this paper, we propose a periodic trajectory generation algorithm to generate trajectories that pass through arbitrary points in the state space and the corresponding criteria for determining the periodic property of the generated trajectories. A novel Lyapunov based control algorithm is proposed to eliminate the difficulties of the LQR based controller. The proposed methods can be applied to a class of UMS and a UMS can be transferred from an initial position to the desired position by tracking the generated periodic trajectories. Such ability is critical for realizing complex tasks and extending the application areas of UMS. The proposed periodic trajectory generation and analysis methods and tracking controller are implemented using an Acrobot prototype and the experimental results demonstrated their performance and verified their effectiveness.