Hybrid dynamical systems are useful for modeling legged locomotion, yet proving existence and regularity of periodic gaits remains difficult. This study analyzes a one-dimensional hybrid reduction of the SLIP model for human running under constant angular velocity during stance, retaining full leg compression and a time-periodic gravitational potential. Stance dynamics include a linear spring with stiffness k, rest length l0, mass m, gravity g, and a sinusoidally varying potential; flight dynamics is parabolic. The resulting system is examined via oscillation theory, Poincaré maps, and intersection theory for conics, then numerically simulated and fitted to experimental data. The main theorem gives necessary and sufficient conditions for existence and multiplicity of 1-periodic smooth solutions continuous across touchdown and takeoff. A one-parameter family of C0 periodic solutions exists for all parameters, while a unique smooth C1 solution occurs only if ω2<k/m and ω2≠k/(2m); no C2 solutions exist. Model fits to human center-of-mass data in treadmill and overground running yield relative L∞L∞ errors below 10%. Results show that high stance frequency relative to leg-spring frequency induces non-smooth, collisional transitions. Despite relaxing small-compression and fixed-roude assumptions, the model yields explicit expressions for biomechanical quantities and accurately captures experimental motion. Biologically, smoothness constraints restrict feasible stiffness–frequency combinations, explaining observed gait patterns and offering guidance for robotic controller design.
