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Traditional geometric mechanics models used in locomotion analysis rely heavily on systems having symmetry in SE(2) (i.e., the dynamics and constraints are invariant with respect to a system’s position and orientation) to simplify motion planning. As a result, the symmetry assumption prevents locomotion analysis on non-flat surfaces because the system dynamics may vary as a function of position and orientation. In this paper, we develop geometric motion planning strategies for a mobile system moving on a position space whose manifold structure is a cylinder: constant non-zero curvature in one dimension and zero curvature in another. To handle this non-flat position space, we adapt conventional geometric mechanics tools - in particular the system connection and the constraint curvature function - to depend on the system orientation. In addition, we introduce a novel constraint projection method to a variational gait optimizer and demonstrate how to design gaits that allow the example system to move on the cylinder with optimal efficiency.