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- A Case Study in Approximate Linearization: The Acrobot Example + (The acrobot is a simple mechanical system … The acrobot is a simple mechanical system patterned after a gymnast</br>performing on a single parallel bar. By swinging her legs, a gymnast</br>is able to bring herself into an inverted position with her center of</br>mass above the part and is able to perform manuevers about this</br>configuration. This report studies the use of nonlinear control</br>techniques for designing a controller to operate in a neighborhood of</br>the manifold of inverted equilibrium points. The techniques described</br>here are of particular interest because the dynamic model of the</br>acrobot violates many of the necessary conditions required to apply</br>current methods in linear and nonlinear control theory.</br><p></br>The approach used in this report is to approximate the system in such</br>a way that the behavior of the system about the manifold of</br>equilibrium points is correctly captured. In particular, we construct</br>an approximating system which agrees with the linearization of the</br>original system on the equilibrium manifold and is full state</br>linearizable. For this class of approximations, controllers can be</br>constructed using recent techniques from differential geometric control </br>theory. We show that application of control laws derived in this</br>manner results in approximate trajectory tracking for the system under</br>certain restrictions on the class of desired trajectories. Simulation</br>results based on a simplified model of the acrobot are included. a simplified model of the acrobot are included.)