This paper presents a geometry-aware predictive control framework for bipedal obstacle crossing across walking and running regimes. The proposed approach integrates a phase-structured trajectory generator with a constrained model predictive control (MPC) tracking layer to generate coordinated pelvis and swing-foot motions during obstacle traversal. The crossing process is represented through five interpretable phases: pre-crossing, leading-leg clearance, straddle, trailing-leg clearance, and recovery. Within this structure, swing-foot elevation is adapted as a function of obstacle height, while lateral stance widening and pelvis repositioning are introduced as functions of obstacle depth to improve transverse clearance and body alignment. A continuous gait-parameter evolution enables smooth walk-to-run transition without discrete controller switching. The generated references are tracked using MPC to regulate motion smoothness, clearance feasibility, touchdown behavior, and lateral admissibility. The framework is evaluated in reduced-order simulation over five progressively difficult obstacles under walking and running conditions. The robot successfully crosses the first four obstacles, achieving minimum body-obstacle clearances of 0.036, 0.036, 0.059, and 0.017 m, respectively. The fifth and most demanding obstacle leads to collision, identifying the current operating boundary of the proposed reduced-order formulation. Overall, the method provides a compact, interpretable, and computationally efficient basis for intelligent obstacle-aware bipedal locomotion.

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