Inverted Pendulum

In order to run the simulation, simply run main.py

In order to save animations ImageMagick is required. It can be installed from here: https://imagemagick.org/script/download.php

Simulation of the pole cart system

The equations of motion that are presented below. Here the pendulum and cart viscous damping is set to zero. I.e. $b_c = b_p = 0$

Cart equation:

$$ \ddot{x}_{cx} = \frac{F_m - b_c\dot{x} + m_pL_p\ddot{\theta} \cos (\theta) - m_pL_p\dot{\theta}^2 \sin (\theta)}{m_c + m_p} $$

Pendulum equation:

$$ \ddot{\theta} = \frac{-b_p\dot\theta + m_pL_pg\sin(\theta) + m_pL_p\ddot{x}_{cx}\cos(\theta)}{ I_p +m_pL_p^2} $$

Swing-up

In order to swing the pendulum up, an energy pumping method is used. $E_t$ is the wanted energy, and $E_p$ is the current pendulum energy.

$$ u(t) = (E_p(t) - E_t)\dot\theta\cos(\theta) $$

Where

$$ E_p(t) = \frac{1}{2}I_p\dot\theta^2 + m_pgl_p\cos(\theta) ~~ and ~~ E_t = m_pgl_p $$

State feedback

The linearized equations of motion, in state space representation, is presented below. The model is applied when calculating the state feedback gain matrix.

$$ \begin{bmatrix} \dot x_{cx} \\ \ddot x_{cx}\\ \dot \theta \\ \ddot \theta \end{bmatrix} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & \frac{-b_c(m_pL_p^2+I_p)}{(L_p^2m_c+I_p)m_p+I_pm_c} & \frac{L_p^2gm_p^2}{(L_p^2m_c+I_p)m_p+I_pm_c} & \frac{-b_pL_pm_p}{(L_p^2m_c+I_p)m_p+I_pm_c} & \\ 0 & 0 & 0 & 1 \\ 0 & \frac{-bm_pL_p}{(L_p^2m_c+I_p)m_p+I_pm_c} & \frac{m_pgL_p(m_c+m_p)}{(L_p^2m_c+I_p)m_p+I_pm_c} & \frac{-b_p(m_c+m_p)}{(L_p^2m_c+I_p)m_p+I_pm_c} \end{bmatrix} \begin{bmatrix} x_{cx} \\ \dot x_{cx}\\ \theta \\ \dot \theta \end{bmatrix} + \begin{bmatrix} 0 \\ \frac{m_pL_p^2+I_p}{(L_p^2m_c+I_p)m_p+I_pm_c}\\ 0 \\ \frac{m_pL_p}{(L_p^2m_c+I_p)m_p+I_pm_c} \end{bmatrix} u(t) $$