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# 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:

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