Blind juggling only works if the paddle is perpendicular to the ball velocity at impact (see the red highlights that indicate velocity and paddle alignment in the sketch). Instead of actively rotating the paddle with a motor, we use a mechanism called a “four bar linkage.” The four bar linkage passively keeps the paddle perpendicular to the ball velocity at impact.
We obtained the link lengths and through an optimization procedure that in addition to keeping the paddle perpendicular to the ball impact velocity, the optimization also tried to match the natural period of the pendulum to the ball flight time. If the period and ball flight time were not matched up, we would need very powerful motors to force the pendulum to swing in phase with the ball.
