Cart-Pole Equations of Motion

For generalized coordinates $q={\left[\begin{array}{cc}s& \theta \end{array}\right]}^{\top }$ the system’s kinetic and potential energy are $$\begin{array}{rl}T& =\frac{1}{2}\left(m_c{\dot{s}}^2+m_p\left({\dot{x}}_p^2+{\dot{y}}_p^2\right)\right)\\ V& =-g{m}_p\cos (\theta ),\end{array}$$ where $$\begin{array}{rl}{\dot{x}}_p& =\dot{s}+l\dot{\theta }\cos (\theta )\\ {\dot{y}}_p& =l\dot{\theta }\sin (\theta ).\end{array}$$ The individual terms of the manipulator equations¹ are then $$\begin{array}{rl}M& =\left[\begin{array}{cc}{m}_c+m_p& m_{p}l\cos \left(\theta \right)\\ m_{p}l\cos \left(\theta \right)& m_p{l}^2\end{array}\right]\\ c& =\left[\begin{array}{c}-m_{p}l\sin \left(\theta \right){\dot{\theta }}^2\\ 0\end{array}\right]\\ {\tau }_g& =\left[\begin{array}{c}0\\ -g{m}_{p}l\sin \left(\theta \right)\end{array}\right]\\ B& =\left[\begin{array}{c}1\\ 0\end{array}\right],\end{array}$$ assuming a single input is acting in the direction of $s$.

Linearization in the upright configuration

From the equations we may see that for the state and input vectors $$\begin{array}{rl}x& =\left[\begin{array}{c}q\\ \dot{q}\end{array}\right]={\left[\begin{array}{cccc}s& \theta & \dot{s}& \dot{\theta }\end{array}\right]}^{\top }\\ u& =\left[\begin{array}{c}u\end{array}\right]\end{array}$$ all points $$\begin{array}{rl}{x}^{\ast }& =\left[\begin{array}{c}{q}^{\ast }\\ {\dot{q}}^{\ast }\end{array}\right]={\left[\begin{array}{cccc}s& \pi & 0& 0\end{array}\right]}^{\top },s\in \mathbb{R}\\ u^{\ast }& =\left[\begin{array}{c}0\end{array}\right]\end{array}$$ are stationary. In this configuration the relevant terms for the system’s linearization are $$\begin{array}{rl}M(q^{\ast })& =\left[\begin{array}{cc}{m}_c+m_p& m_{p}l\\ m_{p}l& m_p{l}^2\end{array}\right]\\ \frac{\partial {\tau }_g}{\partial q}(q^{\ast })& =\left[\begin{array}{cc}0& 0\\ 0& g{m}_{p}l\end{array}\right]\\ \frac{\partial B}{\partial q}(q^{\ast })& =0\\ B(q^{\ast })& =\left[\begin{array}{c}1\\ 0\end{array}\right]\end{array}$$

1. The python script for the derivation of the following terms can be found here. A very similar form of these equations of motion can also be found in section 3.2.1 of Underactuated2023. ↩