Design of Adaptive Controllers based on Christoffel Symbols of First Kind

The present chapter is aimed at systematically exposing the reader to certain modern trends in designing advanced robot controllers. More specifically, it focuses on a new and improved method for building suitable adaptive controllers guaranteeing asymptotic stability. It covers the complete design cycle, while providing detailed insight into most critical design issues of the different building blocks. In this sense, it takes a more global design perspective in jointly examining the design space at control level as well as at the architectural level. The primary purpose is to provide insight and intuition into adaptive controllers based on Christoffel symbols of first kind for a serial-link robot arm, (Mulero-Martinez, 2007a). These controllers are referred to as static since the positional dependence of the nonlinear functions. In this context, the preferred method of nonlinear compensation is the method of building emulators. Often, however, the full power of the method is overlooked, and very few works deal with these techniques at the level of detail that the subject deserves. As a result, the chapter fills that gap and includes the type of information required to help control engineers to apply the method to robot manipulators. Developed in this chapter are several deep connections between dynamics analysis and implementation emphasizing the powerful adaptive methods that emerge when separate techniques from each area are properly assembled in a larger context. After beginning with a comprehensive presentation of the fundamentals of these techniques, the chapter addresses the problem of factorization of the Coriolis/centripetal matrix, (Mulero-Martinez, 2009). This aspect is crucial when designing non-linear compensators by emulation. At this point, it is provided a concise and didactically structured description of the design of emulators as matters stand, (Mulero-Martinez, 2006). Specifically, emulators are split up into sub-emulators to improve and simplify the design of controllers while making faster the updating of parameters. From a practical point of view, the implementation is developed by resorting to parametric structures. This means to obtain a set of system's own function as regression functions. Most of the adaptive schemes start from the notable property of linearity in the parameters, which lead naturally to equivalent structures when designing emulators for the nonlinear terms. When the linearity in the parameters (LIP) is considered as a first assumption in the 10