On generating pre-defined periodic motions in underactuated mechanical systems: the cart-pendulum example

Abstract We study the problem of generating oscillations in underactuated mechanical systems with a chosen time-evolution of some of the generalized coordinates. We consider a classical planar pendulum on a cart example and find conditions of existence of a solution. These conditions are expressed in terms of functions defining synchronization between the actuated and underactuated variables known as virtual holonomic constraints. Explicit expressions for these functions are computed for the cart-pendulum system where the pendulum angle is set to follow a trajectory like a pure sinusoidal waveform around the upright equilibrium. Once the valid virtual constraint has been found with the proposed method, the earlier developed techniques can be applied in order to design an orbitally stabilizing feedback control law.

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