Linearizing Robotic Manipulator's Dynamics Using Koopman Operator and Applying Generalized Predictive Control

Abstract

This paper presents an example of the linearization of a robot manipulator's dynamics using Koopman operators (KP), and presents practical issues to be considered when applying the principal of KP theory to robotic systems. Using the KP, this work presents the successful design of a Generalized Predictive Controller (GPC) that is able to handle any posture/velocity/acceleration for a 3-Dof robot arm, and shows that our GPC performance outperforms a traditional PD controller under a very wide range of operating conditions. KP have only recently begun to attract attention in the field of robotics, and only a few application examples are available. Its basic idea is to lift a non-linear function to a higher dimensional space by extending its explanatory variables, and these new variables are often referred to as “observables”. In this higher dimension space, nonlinear functions can be described in a linear form. Here, how the observables should be augmented depends on the dynamical system's property. For the robotic manipulator, it still remains unclear what is the best set of observables to describe its dynamics. In this study, the optimal set of observables was acquired by expanding the robot dynamics equation and using its monomials. The KP was then identified by the least-squares minimization, and an approximate linear state-space model of the robot was identified. Using this model a GPC controller was designed, and it was shown that our observables choice was properly describing the non-linear behavior globally and made the linear state space model accurate as same as previous research.