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.

We present a real time algorithm for humanoid 3D walking and/or running based on a Model Predictive Control (MPC) approach. The objective is to generate a stable gait that replicates a footstep plan as closely as possible, that is, a sequence of candidate footstep positions and orientations with associated timings. For each footstep, the plan also specifies an associated reference height for the Center of Mass (CoM) and whether the robot should reach the footstep by walking or running. The scheme makes use of the Variable-Height Inverted Pendulum (VH-IP) as a prediction model, generating in real time both a CoM trajectory and adapted footsteps. The VH-IP model relates the position of the CoM to that of the Zero Moment Point (ZMP); to avoid falling, the ZMP must be inside a properly defined support region (a 3D extension of the 2D support polygon) whenever the robot is in contact with the ground. The nonlinearity of the VH-IP is handled by splitting the gait generation into two consecutive stages, both requiring to solve a quadratic program. Thanks to a particular triangular structure of the VH-IP dynamics, the first stage deals with the vertical dynamics using the Ground Reaction Force (GRF) as a decision variable. Using the prediction given by the first stage, the horizontal dynamics become linear time-varying. During the flight phases, the VH-IP collapses to a free-falling mass model. The proposed formulation incorporates constraints in order to maintain physically meaningful values of the GRF, keep the ZMP in the support region during contact phases, and ensure that the adapted footsteps are kinematically realizable. Most importantly, a stability constraint is enforced on the time-varying horizontal dynamics to guarantee a bounded evolution of the CoM with respect to the ZMP. Furthermore, we show how to extend the technique in order to perform running on tilted surfaces. We also describe a simple technique that receives input high-level velocity commands and generates a footstep plan in the form required by the proposed MPC scheme. The algorithm is validated via dynamic simulations on the full-scale humanoid robot HRP-4, as well as experiments on the small-sized robot OP3.

The feasibility region of a Model Predictive Control (MPC) algorithm is the subset of the state space in which the constrained optimization problem to be solved is feasible. In our recent Intrinsically Stable MPC (IS-MPC) method for humanoid gait generation, feasibility means being able to satisfy the dynamic balance condition, the kinematic constraints on footsteps as well as an explicit stability condition. Here, we exploit the feasibility concept to build a step timing adapter that, at each control cycle, modifies the duration of the current step whenever a feasibility loss is imminent due, e.g., to an external perturbation. The proposed approach allows the IS-MPC algorithm to maintain its linearity and adds a negligible computational burden to the overall scheme. Simulations and experimental results where the robot is pushed while walking showcase the performance of the proposed approach.

We propose a novel walking control scheme based on the dynamics of the Linear Inverted Pendulum (LIP) model. The pattern generation incorporates a model of contact forces, enabling closed-loop control of the humanoid robot's state, including the Center of Mass (CoM) position, velocity, and Zero Moment Point (ZMP). No additional control policies are required to maintain static and dynamic balance. Our approach also includes dynamic re-planning of step locations and timings, thus preserving the LIP's boundedness condition. We validated this controller on five different humanoid robots, testing its robustness through various disturbances, including sudden pushes during walking and static phases. Additionally, our controller demonstrated effective locomotion over uneven and compliant terrain. Both simulation and experimental results confirm the effectiveness and robustness of this controller.