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arxiv: 1907.06029 · v1 · pith:JRVZCPMNnew · submitted 2019-07-13 · 📡 eess.SY · cs.SY

Gait Generation using Intrinsically Stable MPC in the Presence of Persistent Disturbances

Filippo M. Smaldone , Nicola Scianca , Valerio Modugno , Leonardo Lanari , Giuseppe Oriolo This is my paper

Pith reviewed 2026-05-24 22:16 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords humanoid locomotionmodel predictive controldisturbance observerbalance maintenancezero moment pointcenter of masspersistent disturbancesquadratic programming
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The pith

A disturbance observer modifies the stability constraint in IS-MPC to keep the center of mass bounded relative to the zero moment point despite persistent disturbances.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends Intrinsically Stable Model Predictive Control to handle persistent disturbances on humanoid robots. It does this by designing a disturbance observer and using its estimate to adjust the stability constraint inside the quadratic programming problem. A sympathetic reader would care because walking robots often face ongoing forces like pushes or slopes that can lead to falls, and this method promises to maintain balance without losing the guarantees of the original approach. The validation comes from simulations on the linear inverted pendulum and a NAO robot in dynamic simulator.

Core claim

By designing a disturbance observer whose estimate is incorporated into a modified stability constraint within the quadratic program formulation, the extended Intrinsically Stable MPC guarantees boundedness of the center of mass with respect to the zero moment point even when persistent disturbances are present.

What carries the argument

The disturbance observer providing an estimate that modifies the stability constraint in the QP optimization.

If this is right

  • The approach applies to both the linear inverted pendulum model and full dynamic simulations of the NAO humanoid.
  • Boundedness of the CoM relative to the ZMP is maintained under persistent disturbances.
  • The method integrates the observer estimate directly into the existing IS-MPC framework without major restructuring.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the observer accuracy holds in practice, this could enable more robust walking on uneven terrain or under wind.
  • Similar observer-based modifications might extend to other stability criteria beyond ZMP.
  • Testing with varying disturbance magnitudes could reveal limits of the estimation accuracy assumption.

Load-bearing premise

The disturbance observer must be tuned to deliver estimates accurate enough that the modified stability constraint still ensures the boundedness property.

What would settle it

A test case where the actual disturbance differs significantly from the observer estimate, leading to the center of mass position becoming unbounded relative to the zero moment point despite using the modified constraint.

Figures

Figures reproduced from arXiv: 1907.06029 by Filippo M. Smaldone, Giuseppe Oriolo, Leonardo Lanari, Nicola Scianca, Valerio Modugno.

Figure 1
Figure 1. Figure 1: A pendulum attached to the robot elbow is used to simulate an [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Balancing in the presence of a known constant force acting on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Gait generation in the presence of a known constant force acting [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Gait generation in the presence of an unknown constant disturbance [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Gait generation in the presence of an unknown rapidly-varying [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Gait generation in the presence of an unknown constant disturbance [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: NAO dynamic simulation in the presence of an unknown constant [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: NAO dynamic simulation in the presence of a slowly-varying force [PITH_FULL_IMAGE:figures/full_fig_p006_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: NAO dynamic simulation in the presence of an oscillating mass [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the authors' prior Intrinsically Stable MPC (IS-MPC) framework, which ensures boundedness of the center of mass (CoM) with respect to the zero moment point (ZMP), to handle persistent disturbances. This is done by designing a disturbance observer and incorporating its estimate into a modified stability constraint within the quadratic programming (QP) formulation of the MPC. The approach is validated through MATLAB simulations on the Linear Inverted Pendulum (LIP) model and dynamic simulations of a NAO humanoid robot in the DART simulator.

Significance. If the disturbance observer provides estimates accurate enough to maintain the stability guarantee, the method offers a way to extend IS-MPC to realistic scenarios with persistent disturbances, potentially improving robustness in humanoid locomotion control. The simulation-based validation on both simplified and full robot models provides initial evidence of practicality.

major comments (2)
  1. [Abstract] Abstract: the central claim that boundedness of the CoM w.r.t. ZMP is guaranteed under persistent disturbances requires that the observer estimation error remain inside a region that does not invalidate the modified stability constraint, but no explicit ultimate bound on that error, no assumption on the disturbance class (constant, harmonic, etc.), and no theorem showing that the closed-loop QP still enforces the intrinsic stability property when the estimate is inexact are provided.
  2. [Abstract] Validation description (abstract): the support for the claim rests on simulations for the LIP and NAO in DART, yet no quantitative metrics (CoM-ZMP deviation, observer error norms), error analysis, or comparison against the nominal IS-MPC under the same disturbances are reported, leaving the practical preservation of the guarantee unverified.
minor comments (2)
  1. [Abstract] Abstract: 'lead the the robot to a fall' contains a repeated word.
  2. [Abstract] Abstract: the description of how the observer estimate modifies the stability constraint is high-level; an equation or explicit statement of the modified constraint would aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback highlighting gaps in the theoretical support for the stability claim and in the quantitative validation. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that boundedness of the CoM w.r.t. ZMP is guaranteed under persistent disturbances requires that the observer estimation error remain inside a region that does not invalidate the modified stability constraint, but no explicit ultimate bound on that error, no assumption on the disturbance class (constant, harmonic, etc.), and no theorem showing that the closed-loop QP still enforces the intrinsic stability property when the estimate is inexact are provided.

    Authors: We agree that the manuscript provides neither an explicit ultimate bound on the observer error, assumptions on the disturbance class, nor a theorem establishing that the QP enforces intrinsic stability under inexact estimates. The current formulation assumes the estimate is sufficiently accurate for the modified constraint to hold, but this is not rigorously analyzed. We will revise the abstract to qualify the boundedness claim and add a dedicated analysis (including disturbance assumptions and error-bound conditions) in the revised manuscript. revision: yes

  2. Referee: [Abstract] Validation description (abstract): the support for the claim rests on simulations for the LIP and NAO in DART, yet no quantitative metrics (CoM-ZMP deviation, observer error norms), error analysis, or comparison against the nominal IS-MPC under the same disturbances are reported, leaving the practical preservation of the guarantee unverified.

    Authors: The referee is correct that the abstract and validation lack quantitative metrics, error norms, and direct comparisons to nominal IS-MPC. While the manuscript presents simulation results on both the LIP and NAO, these are primarily qualitative. We will update the abstract with key quantitative indicators and expand the results section with error analysis and comparisons in the revision. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior IS-MPC; observer-based extension adds independent content

full rationale

The derivation extends the nominal IS-MPC boundedness guarantee (from prior work) by introducing a disturbance observer whose estimate modifies the stability constraint inside the QP. This step is not a redefinition, fit-to-prediction, or self-citation chain; the observer design and modified constraint constitute new content. The self-citation supports only the baseline nominal case and is not load-bearing for the persistent-disturbance claim. No equations reduce by construction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard assumptions from MPC and legged robot dynamics literature plus the ad-hoc design of the observer for this extension; no free parameters or invented entities are specified in the abstract.

axioms (2)
  • domain assumption The linear inverted pendulum (LIP) model sufficiently approximates the humanoid robot dynamics for gait planning under disturbances.
    Validation explicitly uses the LIP model as stated in the abstract.
  • ad hoc to paper A disturbance observer can be designed whose estimate is accurate enough to modify the stability constraint and maintain boundedness.
    This is the core mechanism described for extending IS-MPC to persistent disturbances.

pith-pipeline@v0.9.0 · 5662 in / 1440 out tokens · 41193 ms · 2026-05-24T22:16:08.287425+00:00 · methodology

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Lean theorems connected to this paper

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
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    Relation between the paper passage and the cited Recognition theorem.

    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.

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The paper's claim is directly supported by a theorem in the formal canon.
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The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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The paper appears to rely on the theorem as machinery.
contradicts
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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