The Control of a Three-Link Inverted Pendulum Near the Equilibrium Point

The problem of control of a plane three-link inverted pendulum by means of one or two torques applied at ball joints is considered. This pendulum is an example of a nonlinear underactuated mechanical system, i.e., a system in which the number of degrees of freedom exceeds the dimension of the vector of the generalized controlling force. A three-link pendulum has eight different equilibrium positions, at which some links are directed upwards, and other are directed downwards. All equilibrium positions, except the lower position, are unstable. The pendulum controllability is examined in the linear approximation in the neighborhood of these equilibrium positions for different options of control: by means of external torques applied to some links, or by means of internal torques at ball joints. In the cases of controllability, the control limited by module is constructed in the feedback form to move the pendulum from the neighborhood of the given equilibrium position to the equilibrium position over a finite time period.