Simulation Results

For all the control schemes, the cart-pole simulation was run for 100 times and the number of successful swing-ups was collected. As in [8], a swing-up is considered successful if the final angle is between $-0.133\pi$ and $0.133\pi$, final angular velocity between $-2$   rad$/$s and $2$   rad$/$s, and the cart has not crashed into the boundaries of the area during swing-up. The results of all the simulations are collected in Table II. For each simulation type, the number of successful swing-ups and the number of partial successes are listed. The partial successes include all the simulation runs that at some point reached the desired state, but possibly still failed either because the pole was not stabilised or the cart crashed into a wall.

Figure 4: Example of a successful swing-up with NMPC and low noise. The cart starts from the middle with the pole hanging down, and goes left to swing the pole up.

Table II: Results: Number of successful and semi-successful (in brackets) swing-ups with low and high noise level $\sigma$.

Setting	$\sigma=0.001$		$\sigma=0.1$
Direct Control	14	(48)	4	(31)
Optimistic Inference Control	97	(100)	94	(98)
NMPC	100	(100)	94	(95)
NMPC (only $y$ and $\phi$ observed)	14	(66)	1	(21)
NMPC ( $\mathbf{f}=\mathbf{I}$)	100	(100)	70	(70)
NMPC ( $\mathbf{f}=\mathbf{I}$, only $y$ and $\phi$)	0	(0)	0	(0)