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We present the design of the Blind Juggler: a robot that is able to juggle an unconstrained ball without feedback at heights of up to 2 m. The robot actuates a parabolic aluminum paddle with a linear motor. We achieve open-loop stability of the ball trajectory with two design parameters: 1) the curvature of the parabolic paddle and 2) the acceleration of the paddle at impact. We derive a linear map of perturbations of the nominal ball trajectory over a single bounce and obtain local stability of the trajectory by tuning the eigenvalues of this mapping with the two design parameters. We consider nine ball states in this analysis, including ball spin. Experimental data provide the impact states of the ball and paddle. From these data, we can identify system parameters and infer the process noise introduced into the system. We then combine the experimental noise power spectral densities with a model of the system and optimize the design parameters such that the impact of the process noise on juggling performance is minimized. Theoretical as well as experimental results of the optimization are discussed.