Nonlinear Decoupling Theory with Applications to Robotics (System, Decomposition, Differential Geometry).

Abstract

Some theoretical results on nonlinear decoupling theory are presented and their applications to robotic manipulator control are discussed. First, refinements and extensions of some known results on feedback decoupling of nonlinear systems are given. Precise definitions of decoupling and decomposition are stated. Some conditions under which the two definitions are equivalent for nonlinear systems are found. A previously known condition is shown to be necessary as well as sufficient for a system to be decouplable or locally decomposable. Second, we obtain new results which characterize the whole class of nonlinear feedback control laws which decouple or decompose. These results are important from both mathematical and engineering viewpoints. For instance, there exist systems where our results allow the stable decoupling of a decouplable system, while former results do not. The class of decoupling control laws is characterized by solutions of certain first order partial differential equations. The class of decomposing control laws is characterized by simple feedback laws applied to a st and ard decomposed system (SDS). The SDS is similar to the decomposed system of Isidori, Krener, Gori-Giorgi, and Monaco but has finer structure. These new results are provided by a generalization of ideas used by Gilbert for linear systems. Third, we discuss a form of approximate decoupling. We neglect fast dynamics of a system to obtain a computationally simple control law. It is shown that when the neglected dynamics are sufficiently fast, the simplified law decouples the actual system "approximately" in a certain sense. Finally, these results are applied to decoupled control of robotic manipulators. Two cases are considered. In the first case, actuator dynamics are completely neglected. In the second case, the dynamics of a significant class of actuators are taken into account. Our formulas for the complete class of decoupling control laws unify and generalize previous results on the decoupled control of robotic manipulators. For example, it is possible to achieve decoupled control of the end-effector.