This paper studies the feedback control of a class of complementary-slackness hybrid mechanical systems. Roughly, the systems we study are composed of an uncontrollable part (the "object") and a controlled one (the "robot"),linked by a unilateral constraint and an impact rule. A systematic and general control design method for this class of systems is proposed. The approach is a nontrivial extension of the one degree-of-freedom (DOF) juggler control design. In addition to the robot control, it is also useful to study some intermediate controllability properties of the object's impact Poincare mapping, which generally takes the form of a nonlinear discrete-time system. The force input mainly consists of a family of dead-beat feedback control laws, introduced via a recursive procedure, and exploiting the underlying discrete-time structure of the system. The main goal of this paper is to highlight the role of various physical and control properties characteristic of the system on its stabilizability properties and to propose solutions in certain cases.