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Underactuated mechanical system has less independent inputs than the degrees of freedom(DOF) of the mechanism. The energy efficiency of this class of mechanical systems is an essential problem in practice. On the basis of the sufficient and necessary condition that concludes a single input nonlinear system is differentially flat, 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 relative degree of the system equals to the dimension of the state space. If the flat output of the underactuated system can be solved explicitly, and by constructing a smooth curve with satisfying given boundary conditions in flat output space, an energy efficiency optimization method is proposed for the motion planning of the differentially flat underactuated mechanical systems. The inertia wheel pendulum is used to verify the proposed optimization method, and some numerical simulations show that the presented optimal motion planning method can efficaciously reduce the energy cost for given control tasks.
Underactuated mechanical system has less independent inputs than the degrees of freedom (DOF) of the mechanism. The energy efficiency of this class of mechanical systems is an essential problem in practice. On the basis of the sufficient and necessary condition that concludes a single input nonlinear system is differentially flat, 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 relative degree of the system equals to the dimension of the state space. the underactuated system can be solved explicitly, and by constructing a smooth curve with satisfying given boundary conditions in flat output space, an energy efficiency optimization method is proposed for the motion planning of the differentially flat underactuated mechanical systems. The inertia wheel pendulum is used to verify the proposed optimization method, and some numerical simulations show that the presented Optimal motion planning method can efficaciously reduce the energy cost for given control tasks.