Some New Insights on the Classical Beam and Ball Balancing Experiment

A complete set of nonlinear differential equations of motion is derived for a beam and ball setup. The system obviously has unstable dynamics and may exhibit minimum phase (MP) or non-minimum phase (NMP) behavior depending on the ball radius. The system´s linearized equations are contrasted with different levels of approximations that are often assumed. A transparent interpretation of the NMP behavior is also made in light of the physical setup. Also included is a frequency domain controller design for the NMP unstable case. In particular, a controller is synthesized for keeping the linear and angular position of the ball and the beam within given tolerances under the limited control effort of the actuator.