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:

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

- Dynamic model of an inverted pendulum (part 2)
- Dynamic model of an inverted pendulum (part 3)
- Dynamic model of an inverted pendulum (part 4)
- Dynamic model of an inverted pendulum (part 5)
- Dynamic model of an inverted pendulum (part 6)
- Equation of closed and open loop systems
- PI-based first-order controller

Last update : 02/11/2021