Set-point regulation

Abstract

In the previous chapter we underlined several fundamental properties of EL systems. In particular we saw that the equilibria of an EL plant are determined by the critical points of its potential energy function, moreover the equilibrium is unique and globally stable if this function has a global and unique minimum. We also saw that this equilibrium is asymptotically stable if suitable damping is present in the system. These two fundamental properties motivated Takegaki and Arimoto in [261] to formulate the problem of set point regulation of robots in two steps, first an energy shaping stage where we modify the potential energy of the system in such a way that the “new” potential energy function has a global and unique minimum in the desired equilibrium. Second, a damping injection stage where we now modify the Rayleigh dissipation function. This seminal contribution contained the first clear exposition of the use of energy functions in robotics. (See Subsection 1.3 for a brief review of the literature). It generated a lot of interest in the robotics community since it rigorously established that computationally simple control laws, derived from energy considerations, could accomplish rather sophisticated tasks.