The Cart-Pole and Inverted Pendulum

Balancing a stick on your hand is hard because the stick falls faster the further it tips. A pole on a cart works the same way. It is the classic test problem for control, and the same physics underlies balancing robots, walking, and rockets.

Who balances Time speed Pole length

The dynamics

With cart mass $m_c$, pole mass $m_p$ at length $\ell$, pole angle $\theta$ from upright, and push $F$:

$$\ddot\theta = \frac{-F\cos\theta - m_p \ell\,\dot\theta^2 \sin\theta\cos\theta + (m_c + m_p)\,g\sin\theta}{\ell\,(m_c + m_p\sin^2\theta)}$$

Near upright, $\ddot\theta \approx \frac{(m_c + m_p)\,g}{m_c\,\ell}\,\theta$: the further it tips, the faster it falls. That makes upright an unstable balance point.

LQR

The linear quadratic regulator computes gains $K$ for the state $\mathbf{s} = (x, \dot x, \theta, \dot\theta)$ that minimize a cost on errors and effort. The push is then just $F = -K\mathbf{s}$.

What to notice

Try balancing it yourself in slow motion. To fix a pole leaning right, you must push the cart right, under it, which is the same trick you use with a stick on your hand. LQR does this smoothly and also brings the cart back to the middle. A shorter pole falls faster and is harder to balance, which is why it is easier to balance a broom than a pencil.