This paper presents the design, dynamic modeling, and integration of a single degree of freedom (DOF) robotic leg mechanism intended for tailed quadruped locomotion. The design employs a lightweight six-bar linkage that couples the hip and knee flexion/extension joints mechanically, requiring only a single degree of actuation. By utilizing a parametric optimization, a unique topological arrangement is achieved that results in a foot trajectory that is well suited for dynamic gaits including trot-running, bounding, and galloping. Furthermore, a singular perturbation is introduced to the hybrid dynamic framework to address the lack of robust methods that provide a solution for the differential algebraic equations (DAEs) that characterize closed kinematic chain (CKC) structures as well as the hybrid nature of legged locomotion. By approximating the system dynamics as ordinary differential equations (ODEs) and asymptotically driving the constraint error to zero, CKCs can adopt existing real-time model-based/model-predictive/hybrid-control frameworks. The dynamic model is verified through simulations and the foot trajectory was experimentally validated. Preliminary open-loop planar running demonstrated speeds up to 3.2 m/s. These advantages, accompanied by low-integration costs, warrant this leg as a robust, effective platform for future tailed quadruped research.