A system $($the ball$-$on$-$beam$)$ is governed by the equations of motion:

$(\frac{J}{R^2}+m)r^{}-$mgsin$-mr{}^{}{?}^2=0$

$sin=\frac{d}{L}sin$

Here $m$ is the ball mass, $R$ is the ball radius, $J$ is the ball moment of inertia, $d$ and $L$ are the lengths of the beam and the arm of the actuator. $$ is the beam angle and $$ is the actuator arm angle. The goal is to put the ball at an arbitrary position $(r_r)$ with the beam level