A basic requirement in humanoid gait generation is that the robot maintains dynamic balance while walking. This is certainly true if the Zero Moment Point (ZMP) remains always within the support polygon of the robot. Gait generation schemes typically enforce this condition by computing a suitable trajectory for the robot Center of Mass (CoM). Due to the complexity of the humanoid dynamics, simplified models are used to relate the evolution of the CoM to that of the ZMP, e.g., the Linear Inverted Pendulum (LIP).
A standard tool for generating CoM trajectories that satisfy the dynamic balance constraint is Model Predictive Control (MPC). However, there exists a well-known instability issue when connecting CoM to ZMP trajectories: in general, the CoM trajectory associated to any ZMP profile will include a divergent component. For effective gait generation, it is therefore essential to select a COM trajectory which is stable. However, in standard MPC-based gait generation there is no theoretical guarantee of stability; one simply expects that jerk minimization over a sufficiently long prediction horizon will result in a stable CoM trajectory.
We have developed a scheme that combines the flexibility of MPC with the robustness of guaranteed stability. To this end, we embed in the formulation of the MPC problem an explicit constraint that ensures that the generated CoM trajectory will be stable. We use as motion model a dynamic extension of the LIP, with the ZMP velocity as a control variable. The stability constraint turns out to be linear in this setting, leading to a standard QP problem with equality and inequality constraints. The basic framework for intrinsically stable MPC is developed into a full-fledged gait generation scheme by including automatic footstep placement.
To highlight the main benefit of the proposed method, we have first considered the case of given footsteps for a MATLAB-simulated LIP. As expected, the intrinsically stable MPC shows robust performance with respect to changes in the prediction horizon, whereas standard MPC does not. The complete scheme (including automatic foot placement) has been implemented in V-REP for the NAO humanoid, obtaining effective gaits in various scenarios (variable reference direction, variable reference speed, push recovery). All these results are presented in the following clip.
 N. Scianca, M. Cognetti, D. De Simone, L. Lanari, G. Oriolo, Intrinsically Stable MPC for Humanoid Gait Generation. 16th IEEE-RAS International Conference on Humanoid Robots, Cancún, Mexico, pp. 601-606, 2016 (pdf).