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This paper introduces self-manipulation as a new formal design methodology for legged robots with varying ground interactions. The term denotes a set of modeling choices that permit a uniform and body-centric representation of the equations of motion - essentially a guide to the selection and configuration of coordinate frames. We present the hybrid system kinematics, dynamics, and transitions in the form of a consistently structured representation that simplifies and unites the account of these, otherwise bewilderingly diverse differential algebraic equations. Cleaving as closely as possible to the modeling strategies developed within the mature manipulation literature, self-manipulation models can leverage those insights and results where applicable, while clarifying the fundamental differences. Our primary motivation is not to facilitate numerical simulation but rather to promote design insight. We instantiate the abstract formalism for a simplified model of RHex, and illustrate its utility by applying a variety of analytical and computational techniques to derive new results bearing on behaviors, controllers, and platform design. For each example, we present empirical results documenting the specific benefits of the new insight into the robot's transitions from standing to moving in place and to leaping.