Optimal motion planning for differentially flat underactuated mechanical systems

It is shown that the flat output of the single input underactuated mechanical system can be obtained by finding a smooth output function such that the system has relative degree equals to the dimension of the state space. Assuming the flat output of an underactuated system can be solved explicitly, an optimization method is proposed for the motion planning of the differentially flat underactuated mechanical systems by constructing a shape adjustable curve with satisfying specific boundary conditions in flat output space. The inertia wheel pendulum is used to verify the proposed optimization method and some numerical simulation results are included.