The transformable wheel mechanism of a tri-footed mobile service robot determines its driving performance. Currently, two independent actuators adjust the wheel's three feet adaptively, but we aim to develop novel single-actuator-operable 1-DOF transformable wheel mechanisms. Due to the unexplored nature of 1-DOF mechanisms, the trial-and-error approach may fail to find such mechanisms. Herein, we propose a fully automated gradient-based design approach that simultaneously determines the mechanism's topology, dimension, and joint type without a baseline mechanism. We take both fundamental kinematic and functional (such as torque-force transmission ratio) requirements into account. Our formulation employing a unified ground synthesis model is newly established to consider the trajectory and orientation of the mechanism end-effector, control the number of synthesized prismatic joints, and satisfy the torque-force transmission ratio. Under various design constraints, we successfully synthesized novel mechanisms, and their functionalities were validated via simulations and the motions of their prototypes.