This paper presents a real-time control-constrained Differential Dynamic Programming (DDP) framework for underactuated legged robots. To address the limitation of classical DDP in handling control constraints, we propose an Accelerated Projected Gradient (APG)-based control-constrained DDP (ABC-DDP), which efficiently computes constrained solutions and identifies active sets without repeated Karush-Kuhn-Tucker (KKT) inversions. A virtual constraint is introduced to integrate control constraints within a feasibility-driven multiple-shooting framework, enabling stable optimization even from dynamically infeasible initializations. The proposed method supports real-time model predictive control (MPC) with short horizons under strong underactuation. Simulation results demonstrate static two-leg standing under external disturbances, along with diverse dynamic motions including slow catwalk, upright walking, and high-speed running within a unified MPC framework. To the best of our knowledge, this is the first demonstration of static two-leg standing of a quadruped robot achieved using real-time finite-horizon MPC.

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