Ball on a beam: stabilization under saturated input control with large basin of attraction

This article is devoted to the stabilization of two underactuated planar systems, the well-known straight beam-and-ball system and an original circular beam-and-ball system. The feedback control for each system is designed, using the Jordan form of its model, linearized near the unstable equilibrium. The limits on the voltage, fed to the motor, are taken into account explicitly. The straight beam-and-ball system has one unstable mode in the motion near the equilibrium point. The proposed control law ensures that the basin of attraction coincides with the controllability domain. The circular beam-and-ball system has two unstable modes near the equilibrium point. Therefore, this device, never considered in the past, is much more difficult to control than the straight beam-and-ball system. The main contribution is to propose a simple new control law, which ensures by adjusting its gain parameters that the basin of attraction arbitrarily can approach the controllability domain for the linear case. For both nonlinear systems, simulation results are presented to illustrate the efficiency of the designed nonlinear control laws and to determine the basin of attraction

On optimal swinging of the biped arms

Abstract — A ballistic walking gait is designed for a planar biped with two identical two-link legs, a trunk and two onelink arms. This seven-link biped is controlled via impulsive torques at the instantaneous double support to obtain a cyclic gait. These impulsive torques are applied in six inter-link joints. Then infinity of solutions exists to find the impulsive torques. An energy cost functional of these impulsive torques is calculated to choose a unique solution by its minimization. Numerical results show that for a given time period of the walking gait step and a length of the step, there exists an optimal swinging amplitude of arms. For this optimal motion of the arms, mentioned above cost functional is minimum. I