This paper serves as an extended abstract for a longer paper that contains proofs and a more thorough discussion of the methodology. In this work, the notion of a control-affine system is abstracted to its geometric essence: an affine subbundle. In this abstract setting, notions of control systems and controllability are presented, and general second-order conditions for local controllability are given. The conditions are notable in that their hypotheses involve only the affine subbundle, and objects directly related to it. In this way the conditions we give are guaranteed to be independent of feedback transformations.

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