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Unformatted text preview: ECE 594D Robot Locomotion Winter 2010 Cart-pole system: Equations of motion Nonlinear Dynamics This document provides a derivation of the equations of motion (EOM) for the cart-pole system. The (true) nonlinear dynamic equations are derived first, using a Lagrangian approach; then the system is lin- earized about the upright equilibrium (“inverted pendulum”) position. Figure 1 shows the system. There are two degrees of freedom: the position of the cart, x , and the angle of the pendulum, θ . The system is underactuated, since there is only one actuation: a force, F x , applied on the cart. m p g M θ x Figure 1: Cart-pole system. To begin, we will derive the simple relationships giving the position and velocity of the pendulum: x p =- L sin θ (1) ˙ x p =- L cos θ ˙ θ + ˙ x (2) y p = L cos θ (3) ˙ y p =- L sin θ ˙ θ (4) The “Lagrangian” for a dynamic system is defined as: L = T *- V (5) where T * is the kinetic energy and V is the potential energy . For the cart-pole system:....
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This note was uploaded on 12/29/2011 for the course ECE 594d taught by Professor Teel,a during the Fall '08 term at UCSB.
- Fall '08