  We present IS-MPC, an intrinsically stable MPC framework for humanoid gait
generation which incorporates an explicit stability constraint in the
formulation. The proposed method uses as prediction model a dynamically
extended LIP where ZMP velocities are the control inputs, producing in real
time a gait (including footsteps with the associated timing) that realizes
omnidirectional motion commands coming from an external source. The stability
constraint links the future ZMP velocities to the current system state so as to
guarantee the essential requirement that the generated CoM trajectory is
bounded with respect to the ZMP trajectory. Since the control horizon of the
MPC algorithm is finite, only part of the future ZMP velocities are decision
variables of the QP problem; the remaining part, called tail, must be either
conjectured or anticipated using preview information on the reference motion.
Several possible options for the tail are discussed, and each of them is shown
to correspond to a specific terminal constraint. A theoretical analysis of the
feasibility of the generic MPC iteration is developed and used to obtain
sufficient conditions for recursive feasibility. Finally, it is proved that
IS-MPC guarantees stability of the CoM/ZMP dynamics if it is recursively
feasible. Simulation and experimental results on the NAO and the HRP-4
humanoids are presented to illustrate the performance of the proposed method.
