In this article, we present an intrinsically stable Model Predictive Control (IS-MPC) framework for humanoid gait generation that incorporates a stability constraint in the formulation. The method uses as prediction model a dynamically extended Linear Inverted Pendulum with Zero Moment Point (ZMP) velocities as control inputs, producing in real time a gait (including footsteps with timing) that realizes omnidirectional motion commands coming from an external source. The stability constraint links future ZMP velocities to the current state so as to guarantee that the generated Center of Mass (CoM) trajectory is bounded with respect to the ZMP trajectory. Being the MPC control horizon finite, only part of the future ZMP velocities are decision variables; the remaining part, called tail, must be either conjectured or anticipated using preview information on the reference motion. Several options for the tail are discussed, each corresponding to a specific terminal constraint. A feasibility analysis of the generic MPC iteration is developed and used to obtain sufficient conditions for recursive feasibility. Finally, we prove that recursive feasibility guarantees stability of the CoM/ZMP dynamics. Simulation and experimental results on NAO and HRP-4 are presented to highlight the performance of IS-MPC.

The exact eigenfunctions for the slewing link are found, taking into account a rotating inertia at the base and a payload at the tip. These derive from two equivalent formulations (pseudoclamped and pseudopinned) of the boundary value problem relative to the flexible slewing beam. The exactness of the solution makes it possible to prove the equivalence of these two approaches, which differ in the choice of the noninertial rotating frame. The two related dynamic linear models are then found, and a change of coordinates is given. Experimental measurements validate the theoretical results

Capturability analysis of the linear inverted pendulum (LIP) model enabled walking with constrained height based on the capture point. In this paper, we generalize this analysis to the variable-height inverted pendulum (VHIP) and show how it enables 3-D walking over uneven terrains based on capture inputs. Thanks to a tailored optimization scheme, we can compute these inputs fast enough for real-time model predictive control. We implement this approach as open-source software and demonstrate it in dynamic simulations.