This paper introduces the novel notion of kinematic reductions for mechanical systems and studies their controllability properties. We focus on the class of simple mechanical control systems with constraints and model them as affine connection control systems. For these systems, a kinematic reduction is a driftless control system whose controlled trajectories are also solutions to the full dynamic model under appropriate controls. We present a comprehensive treatment of local controllability properties of mechanical systems and their kinematic reductions. Remarkably, a number of interesting reduction and controllability conditions can be characterized in terms of a certain vector-valued quadratic form. We conclude with a catalog of example systems and their kinematic reductions.

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