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 fiat 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.
The multi-modes feature, the measure of the manipulating flexibility, andself-reconfiguration control method of the underactuated redundant manipulators are investigatedbased on the optimizing technology. The relationship between the configuration of the joint spaceand the manipulating flexibility of the underactuated redundant manipulator is analyzed, a newmeasure of manipulating flexibility ellipsoid for the underactuated redundant manipulator withpassive joints in locked mode is proposed, which can be used to get the optimal configuration forthe realization of the self-reconfiguration control. Furthermore, a time-varying nonlinear controlmethod based on harmonic inputs is suggested for fulfilling the self-reconfiguration. A simulationexample of a three-DOFs underactuated manipulator with one passive joint features some aspects ofthe investigations.