Addressing the challenge of quadrupedal terrain traversal in complex environments with constrained obstacles, this work presents an integrated planning-control framework synergizing trajectory optimization (TO), nonlinear model predictive control (NMPC), and whole-body control (WBC). The core theoretical contribution of this work is to propose an enhanced multiple-shooting (MS) based constrained differential dynamic programming (DDP) solver for kinodynamic NMPC. A novel derivation is proposed that accounts for the defect between shooting segments during the DDP backward pass, which improves traditional single-shooting (SS) based DDP robustness to initial guesses. Furthermore, the NMPC is integrated with high-level TO and low-level WBC to form a unified optimal trajectory planning and control framework, enabling quadruped robots to execute high-speed dynamic motions with extended flight phases through constrained obstacles, which constitutes the main technical contribution of this research. Experimental validation via high-fidelity simulations and hardware implementation on our self-developed platform ArcDog confirms both the solver’s superiority and the efficacy of the framework.