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We present two-dimensional monopods that are capable of stable periodic hopping motions without any feedback. It is shown that a large foot radius is not necessary for self-stabilization and that even one-legged robots with a point foot can perform open-loop stable motions although they have no statically stable standing position. The concept of open-loop stability implies that the actuators receive purely periodic torque or force inputs that are never altered with any feedback to prevent the robot from falling. The robots consist of a toroidal body and a leg that are connected by a hinge powered by a torque. The leg bounces on an adjustable spring. The design of these robots and the choice of model parameter values leading to stable motions is a difficult task that has been accomplished using newly developed stability optimization methods.