Dynamic model of an inverted pendulum (part 1)

Introduction

This post presents the model of an inverted pendulum. This device is composed of an activated trolley and a pendulum which has its center of mass above its pivot point.

In this post, we will assume the following hypotheses:

$m_1$ is the mass of the trolley.
$m_2$ is the mass of the pendulum.
$I_1$ is the moment of inertia of the trolley.
$I_2$ is the moment of inertia of the pendulum.
$L$ is the distance between the pivot and the center of mass of the pendulum.
$\Theta$ is the angle of inclination of the pendulum.
$x_1$ and $y_1$ are the coordinates of the center of mass $C_1$ of the trolley .
$\Theta_1$ is the orientation of the trolley.
$x_2$ and $y_2$ are the coordinates of the center of mass $C_2$ of the pendulum.
$\Theta_2$ is the orientation of the pendulum.

The following notation will be used in this article to avoid too complex equations for the derivatives:

$$ \dfrac{d\Theta}{dt} = \dot{\Theta} $$

And for the second derivative:

$$ \dfrac{d^2\Theta}{dt^2} = \ddot{\Theta} $$

Part 2

Dynamic model of an inverted pendulum (part 1)

Introduction

See also