Lagrange Equation Lab Report

CHAPTER 1: INTRODUCTION

Inverted pendulum systems represent a significant group of mechanical systems used in control education with a number of practical applications. It is evident that the stabilization of a walking human or robot, a launching missile or the vertical movement of a shoulder or arm can all be modeled by some kind of an inverted pendulum system. The diversity of modeled systems is reflected in the variety of available inverted pendulum models. These may differ by:

• the type of actuating mechanism – the system base is moving either in a single axis (classical pendulum system) or in a plane (rotary pendulum system)

• the number of pendulum links attached to the mechanism – single and double pendulum systems are common control

Usually the kinetic energy is a quadratic form of generalities velocities, thus the Lagrange equations (2.13) are ordinary differential equations of order two with respect to generalized coordinates qi , i = 1, 2, . . . , n. Putting initial conditions for qi = qi (t0 ) and q˙ i = q˙ i (t0 ) into the general solution of Lagrange equations we choose the particular solution of the Lagrange equation, that realize the dynamics of considered by us mechanical system. Knowledge of equations of motion and initial conditions allow with absolute certainty reproduce past and determine future of the motion of the considered system. In Lagrangian mechanics 2n variables describing evolution in time of the system are generalized position qi and generalize velocity q˙i . The equations governed the motion are the Lagrange equations of the second kind. This movement takes place in n-dimensional configuration space, with coordinates q1, q2, . . . , qn

To visualise the dynamics using the Poincaré sections we eliminate variable p1 using the conservation of energy low H = E = constant; where H is given by (3.4), we obtain

(3.8)

We must remember that the expression under radical in Eq.(3.8) cannot be negative. Therefore, for given constant parameters m1 , m2, l1 , l2, E, admissible values for the angles θ1, θ2 and momentum p2 subject to some limitations. This limitation causes that picture on Poincaré section possesses boundary.

We will analyse the dynamics of the considered system for the following constant parameters:

m1 = 3, m2 = 1, l1 = 2, l2 = 1, g = 1
Fixing values (θ1, θ2, p1 , p2) = (0, 0, 0, 0) into Eq.(3.4), we can easy to check that the energy minimum corresponding to a state of rest for the pendulum is equal to E0 = −9.
Therefore, with slightly increasing the energy E = −9 + 0.05 we expect that the pendulum will oscillate close to the equilibrium point. Fig. 4.2 represents the (θ2 , p2) plane with chosen the Poincare surface of section θ1 = 0, and p1 > 0 for 50 different initial