Bouncing an Unconstrained Ball in Three Dimensions with a Blind Juggling Robot

We describe the design of a juggling robot that is able to vertically bounce a completely unconstrained ball without any sensing. The robot consists of a linear motor actuating a machined aluminum paddle. The curvature of this paddle keeps the ball from falling off while the apex height of the ball is stabilized by decelerating the paddle at impact. We analyze the mapping of perturbations of the nominal trajectory over a single bounce to determine the design parameters that stabilize the system. The first robot prototype confirms the results from the stability analysis and exhibits substantial robustness to perturbations in the horizontal degree of freedoms. We then measure the performance of the robot and characterize the noise introduced into the system as white noise. This allows us to refine the design parameters by minimizing the H
2
norm of an input-output representation of the system. Finally, we design an H
2
optimal controller for the apex height using impact time measurements as feedback and show that the closed-loop performance is only marginally better than what is achieved with open-loop control.