A remark on Controlled Lagrangian approach

A new approach to trajectory planning for underactuated mechanical systems is presented and discussed based on analysis of feasible behaviors of a standard 2-DOF benchmark example — the cart–pendulum system. Following the Controlled Lagrangians approach of Bloch et al. (2000) [7], we present and re-establish known conditions and forms of feedback control laws for this example, which are leading to an equivalent completely integrable closed-loop Euler–Lagrange system; and then extend them. As shown, full integrability and, in particular, the presence of a linear in velocities first integral of dynamics plays the key role in an elegant new procedure for trajectory planning.