Cart-Pole Swing-Up Task

The Cart-Pole system Kimura99 is a classic benchmark for nonlinear control. The system consist of a pole (which acts as an inverted pendulum) attached to a cart (Figure 1). The force applied to the cart can be controlled, and the goal is to swing the pole to an upward position and stabilise it. This must be accomplished without the cart crashing into the walls of the track.

The observed variables of the system are the position of the cart $s$, angle of the pole measured from the upward position $\phi$, and their first derivatives $s'$ and $\phi'$. Control input is the force $F$ applied to the cart. The detailed dynamics and constraints for the simulated cart-pole system can be found in Kimura99.

Figure 1: The cart-pole system

[cc][cc]$\phi$

A discrete system was simulated with a time step of $\Delta t=0.05$   s. The possible force was constrained between $-10$   N and $10$   N, and the position between $-3$   m and $3$   m. The system was initialised to a random state taken from the uniform distributions $s=[-1,1]$, $s'=[-2,2]$, $\phi=[\pi-1,\pi+1]$, $\phi'=[-3,3]$.