# MPC for Humanoid Gait Generation: Stability and Feasibility

## TL;DR

This paper introduces IS-MPC, a stable model predictive control framework for humanoid gait generation that guarantees stability and feasibility through explicit constraints, validated by simulations and real robot experiments.

## Contribution

It proposes a novel stable MPC formulation with explicit stability constraints for humanoid gait generation, ensuring bounded CoM trajectories and recursive feasibility.

## Key findings

- Guarantees stability of CoM/ZMP dynamics.
- Ensures recursive feasibility of the MPC algorithm.
- Validated on NAO and HRP-4 humanoid robots.

## Abstract

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.