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MPC for Humanoid Gait Generation: Stability and Feasibility

Nicola Scianca, Daniele De Simone, Leonardo Lanari, Giuseppe Oriolo

TL;DR

IS-MPC delivers intrinsically stable humanoid gait generation by embedding a stability constraint that bounds the CoM with respect to the ZMP under a dynamically extended LIP model. It introduces tail-based methods (truncated, periodic, anticipative) to handle unknown ZMP evolution beyond the control horizon, with each tail corresponding to a terminal constraint; anticipative tails are especially effective given preview information. The authors derive sufficient conditions for recursive feasibility and prove that, once recursion is achieved, the CoM/ZMP dynamics are internally stable. The framework is validated through simulations and experiments on NAO and HRP-4, demonstrating feasibility, stability, and real-time performance, with anticipative tails enabling robust operation under varied gaits and previews.

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.

MPC for Humanoid Gait Generation: Stability and Feasibility

TL;DR

IS-MPC delivers intrinsically stable humanoid gait generation by embedding a stability constraint that bounds the CoM with respect to the ZMP under a dynamically extended LIP model. It introduces tail-based methods (truncated, periodic, anticipative) to handle unknown ZMP evolution beyond the control horizon, with each tail corresponding to a terminal constraint; anticipative tails are especially effective given preview information. The authors derive sufficient conditions for recursive feasibility and prove that, once recursion is achieved, the CoM/ZMP dynamics are internally stable. The framework is validated through simulations and experiments on NAO and HRP-4, demonstrating feasibility, stability, and real-time performance, with anticipative tails enabling robust operation under varied gaits and previews.

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.

Paper Structure

This paper contains 29 sections, 6 theorems, 56 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Problem and Approach
  3. Candidate Footstep Generation
  4. Candidate Footstep Timing
  5. Candidate Footstep Placement
  6. IS-MPC: Prediction Model and Constraints
  7. Prediction Model
  8. ZMP Constraints
  9. Kinematic Constraints
  10. IS-MPC: Enforcing Stability
  11. Stability Constraint
  12. Tails
  13. Truncated Tail
  14. Periodic Tail
  15. Anticipative Tail
  16. ...and 14 more sections

Key Result

Proposition 1

For the piecewise-linear $x_z$ in (eq:LinearZ), condition (eq:bcConstr) becomes while (eq:TC) takes the form

Figures (18)

  • Figure 1: A block scheme of the proposed MPC-based framework for gait generation.
  • Figure 2: The proposed rule for determining the step duration $T_s$ as a function of the magnitude $v$ of the reference Cartesian velocity. For comparison, also shown are the rules yielding constant step duration and constant step length.
  • Figure 3: Candidate footsteps generated by the proposed method for different high-level reference velocities corresponding to a circular walk (top), L-walk (center), diagonal walk (bottom). The paths in black are obtained by integrating model (\ref{['eq:OmnidirectionalModel']}) under the reference velocities. Footstep in magenta and cyan refer respectively to the left and right foot.
  • Figure 4: The LIP in the $x$ direction.
  • Figure 5: At time $t_k$, the control variables determined by IS-MPC are the piecewise-constant ZMP velocities over the control horizon. The ZMP velocities after the control horizon are instead conjectured in order to build the tail (see Sect. \ref{['sect:Tails']}). Also shown are the $F$ footstep timestamps placed by the footstep generation module in the preview horizon; $F'$ of them fall in the control horizon.
  • ...and 13 more figures

Theorems & Definitions (6)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Proposition 6