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This paper studies a class of hybrid mechanical systems that locomote by switching between constraints defining different dynamic regimes. We develop a geometric framework for modeling smooth phenomena such as inertial forces, holonomic and nonholonomic constraints, as well as discrete features such as transitions between smooth dynamic regimes through plastic and elastic impacts. We focus on devices that are able to switch between constraints at an arbitrary point in the configuration space. This class of hybrid mechanical control systems can be described in terms of affine connections and jump transition maps that are linear in the velocity. We investigate two notions of local controllability, the equilibrium and kinematic controllability, and provide sufficient conditions for each of them. The tests rely on the assumption of zero velocity switches. We illustrate the modeling framework and the controllability tests on a planar sliding, clamped, and rolling device. In particular, we show how the analysis can be used for motion planning.