Before revision:  MHPC is formulated as a multi-phase receding-horizon Trajectory Optimization (TO) problem, and is solved by an efficient solvercalled Hybrid Systems Differential Dynamic Programming (HSDDP) . MHPC is benchmarked in simulation on a quadruped, a biped, and a quadrotor, demonstrating control performance on par or exceeding whole-body MPC while maintaining a lower computational cost in each case.
After revision:  MHPC is formulated as a multi-phase receding-horizon Trajectory Optimization (TO) problem, and can be implemented using any general multi-phase TO solver . MHPC is benchmarked in simulation on a quadruped, a biped, and a quadrotor, demonstrating control performance on par or exceeding whole-body MPC while maintaining a lower computational cost in each case.