Nonlinear Geometric Control Research Articles

Feedback linearization and linear stabilization control of flexible robots

Abstract An approach to the nonlinear control of flexible robot arms based on the differential geometric nonlinear control theory and the linear quadratic regulator (LQR) theory is considered using only the actuators at the joints. The control is separated into two parts: one derived by the differential geometric structure algorithm is a global external linearization and decoupling nonlinear feedback law for nominal trajectory tracking, and the other synthesized by combined LQR and servocompensator techniques is a robust linear time‐invariant stabilization and error correcting feedback control designed for active damping of elastic vibrations and for robustness to uncertainty in system parameters. By suitable selection of output variables, the suggested control design methodology can be used for either joint space control or task space control. For illustrations, a weightless three‐link PUMA type robot arm with a special flexible third link is tested via simulation.

A nonlinear observer for elastic robots

Robot manipulators in which the joints exhibit a certain amount of elasticity are discussed. The model of such robots is highly nonlinear, and available techniques to build nonlinear observers generally do not apply. A procedure to construct approximate nonlinear observers is proposed that uses a recent approach to deriving observers based on geometric nonlinear control theory together with an approximation technique. The conditions under which an exact observer exists are derived. Some examples are given for which simulation tests assure a good performance even when the observer is connected to a control system.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models

In this paper, we study the attitude control problem for spacecraft with gas jet or momentum exchange actuators, using the recent nonlinear geometric control theory. We give necessary and sufficient conditions for controllability of the system in the case that the gas jet actuators yield one, two, or three independent torques. In the case of momentum exchange devices, controllability is studied with three independent actuators, and controllability is shown to be impossible with fewer devices. The former conditions with gas jet actuators are presented in three equivalent ways, and an equivalence is established with an earlier condition by Baillieul. The local controllability problem is also studied in the case of gas jet actuators yielding two independent torques. Using these results, an algorithm stabilizing the controllable system around an equilibrium state and trajectory is outlined, as proposed by Hermes. In the situations considered, however, the linearized systems are not controllable.

Linear-model-following control and feedback-equivalence to linear controllable systems†

The sufficient conditions developed in the area of model reference adaptive control which guarantee the existence of a linear-model-following control for a non-linear plant are given a geometric interpretation. They are compared with recent results, developed in the area of non-linear geometric control, on global feedback equivalence to linear controllable systems. The comparison gives more insight into the model-following control problem and suggests the generalization of the usual control scheme to a more convenient one, which admits solutions for a larger class of plants.

An example of a nonlinear regulator

The purpose of this note is to use the concept of feedback equivalence to linear controllable systems in order to design a nonlinear regulator for a synchronous generator connected to an infinite bus. The design of the regulator will show how recently developed theorems in the area of geometric nonlinear control theory approach a power system problem, offering an interesting solution.