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.

We present a novel MPC method for humanoid gait generation that is guaranteed to produce stable CoM trajectories. This is obtained by using a dynamic extension of the LIP as motion model, with the ZMP velocity as a control variable, and embedding an explicit stability constraint in the formulation. Such constraint turns out to be linear in the control variables, leading to a standard QP problem with equality and inequality constraints. The intrinsically stable MPC framework is developed into a full-fledged gait generation scheme by including automatic footstep placement. Simulations show that the proposed method is very effective and performs robustly in the presence of changes in the prediction horizon.

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Maintaining balance while walking is not a simple task for a humanoid robot because of its complex dynamics. The presence of a persistent disturbance makes this task even more challenging, as it can cause a loss of balance and ultimately lead the the robot to a fall. In this paper, we extend our previously proposed Intrinsically Stable MPC (IS-MPC), which guarantees boundedness of the CoM with respect to the ZMP, to the case of persistent disturbances. This is achieved by designing a disturbance observer whose estimate is used to compute a modified stability constraint included in the QP problem formulation. The method is validated by MATLAB simulations for the LIP as well as dynamic simulations for a NAO humanoid in DART.

We consider the problem of gait generation for a humanoid robot that must walk to an assigned Cartesian goal. As a first solution, we consider a rather straightforward adaptation of our previous work: an external block produces high-level velocities, which are then tracked by a double-stage intrinsically stable MPC scheme where the orientation of the footsteps is chosen before determining their location and the CoM trajectory. The second solution, which represents the main contribution of the paper, is conceptually different: no highlevel velocity is generated, and footstep orientations are chosen at the same time of the other decision variables in a singlestage MPC. This is made possible by carefully redesigning the motion constraints so as to preserve linearity. Preliminary results on a simulated NAO confirm that the single-stage method outperforms the conventional double-stage scheme.

We consider a pursuit-evasion problem between humanoids. In our scenario, the pursuer enters the safety area of the evader headed for collision, while the latter executes a fast evasive motion. Control schemes are designed for both the pursuer and the evader. They are structurally identical, although the objectives are different: the pursuer tries to align its direction of motion with the line-of-sight to the evader, whereas the evader tries to move in a direction orthogonal to the line-of-sight to the pursuer. At the core of the control scheme is a maneuver planning module which makes use of closedform expressions exclusively. This allows its use in a replanning framework, where each robot updates its motion plan upon completion of a step to account for the perceived motion of the other. Simulation and experimental results on NAO humanoids reveal an interesting asymptotic behavior which was predicted using unicycle as template models for trajectory generation.

We consider a pursuit-evasion problem between humanoids in the presence of obstacles. In our scenario, the pursuer enters the safety area of the evader headed for collision, while the latter executes a fast evasive motion. Control schemes are designed for both the pursuer and the evader. They are structurally identical, although the objectives are different: the pursuer tries to align its direction of motion with the lineof-sight to the evader, whereas the evader tries to move in a direction orthogonal to the line-of-sight to the pursuer. At the core of the control architecture is a Model Predictive Control scheme for generating a stable gait. This allows for the inclusion of workspace obstacles, which we take into account at two levels: during the determination of the footsteps orientation and as an explicit MPC constraint. We illustrate the results with simulations on NAO humanoids.