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

Gait optimization and energetics of ballistic walking for an underactuated biped with knees

In this paper, we study gait optimization of ballistic walking in order to understand the natural dynamics of an underactuated biped with knees. We also propose applications for our understandings. Our optimization problem is solved by fixing energy levels, and then, we attempt to explain how optimal gaits are formed by examining the role of each joint in speeding up. In addition, we explain some natural characteristics of walking. Based on the results, we propose a new cost function to generate various walking gaits, including the optimum. Finally, we evaluate and discuss the energy efficiency of our ballistic walker and other bipedal walkers including humans.clos