Before revision:  We prove that the generated gaits are feasible solutions of the MPC optimization problems. Thus our proposed gait synthesizer works as a fast MPC-style planner to guarantee the feasibility and stability of the full dimensional robot .
After revision:  We prove that the constructed MPC problem can ensure the uniform ultimate boundedness (UUB) of the CoM states and show that our proposed gait synthesizer works as a fast MPC-style planner to guarantee the feasibility and stability of the full dimensional robot .