Sensitivity of Legged Balance Control to Uncertainties and Sampling Period

Johannes Englsberger; Nahuel Villa; Pierre-Brice Wieber (BIPOP)

arxiv: 1907.01805 · v1 · pith:ZK6J3NZ3new · submitted 2019-07-03 · 💻 cs.RO

Sensitivity of Legged Balance Control to Uncertainties and Sampling Period

Nahuel Villa , Johannes Englsberger , Pierre-Brice Wieber (BIPOP) This is my paper

Pith reviewed 2026-05-25 10:16 UTC · model grok-4.3

classification 💻 cs.RO

keywords legged robotsbalance controlrobust controlsampling periodsensor uncertaintiesactuator uncertaintiescenter of masscenter of pressure

0 comments

The pith

Sampling periods as long as 200 ms produce no increase in maximum tracking error for legged balance under bounded sensor and actuator uncertainties.

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

The paper applies robust control theory to bound the worst-case effects of sensor and actuator uncertainties on center-of-mass and center-of-pressure trajectories in legged robots. Because the uncertainties are treated as arbitrary values inside fixed intervals, the resulting feedback gains deliver explicit performance guarantees that hold for every possible realization inside those intervals. The analysis reveals that lengthening the sampling period up to 200 ms leaves the guaranteed maximum tracking error unchanged, thereby preserving the safety margin needed to keep the center of pressure inside the support polygon. This conclusion is obtained by direct computation of the closed-loop response under the robust framework and is checked in both simulation and hardware experiments on a torque-controlled humanoid.

Core claim

When uncertainties in sensors and actuators are allowed to take any value inside prescribed bounds, robust-control synthesis yields feedback gains such that the maximum tracking error on the center of mass and center of pressure remains independent of sampling periods up to 200 ms; consequently the guarantee that balance can be maintained inside a limited support polygon is likewise unaffected by these longer intervals.

What carries the argument

Robust-control synthesis that computes explicit upper bounds on center-of-mass and center-of-pressure tracking error for all uncertainty values inside given intervals.

If this is right

Feedback gains can be chosen once from the robust synthesis and then used at sampling rates as low as 5 Hz without degrading the balance-safety margin.
The same derivations apply without change to both biped and quadruped platforms.
Hardware implementations can reduce control-loop frequency and therefore computational load while retaining the same worst-case performance certificate.
Validation on the Toro humanoid confirms that the predicted error bounds are not exceeded in practice.

Where Pith is reading between the lines

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

Designers could deliberately lower actuator bandwidth or sensor update rates to reduce cost or power without sacrificing the balance guarantee.
The same bounded-uncertainty model could be reused to certify other legged tasks such as walking or push recovery.
If the actual uncertainty ranges turn out narrower than assumed, the synthesis could be rerun to permit even longer sampling periods.

Load-bearing premise

All sensor and actuator errors stay inside the fixed numerical bounds used to set up the robust-control problem.

What would settle it

An experiment that records center-of-mass tracking error exceeding the computed robust bound, or loss of balance, while sampling at 200 ms under sensor and actuator errors that remain inside the assumed intervals.

Figures

Figures reproduced from arXiv: 1907.01805 by Johannes Englsberger, Nahuel Villa, Pierre-Brice Wieber (BIPOP).

**Figure 2.** Figure 2: Following Jury’s simplified stability criterion, th [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: Span of the CoP tracking error p˜span produced by model and estimation errors with span nˆspan = ξˆspan = 1 cm, using the optimal feedback gains k = 2 and λ = ω−1 (ω ≈ 3.2 s−1 for Toro) for different sampling periods τ. The tracking error degrades sharply when τ > ω−1 ln 2 = 216 ms, but it doesn’t improve for sampling periods below this value. VII. EXPERIMENTAL RESULTS The CP linear feedback (45) is implem… view at source ↗

**Figure 5.** Figure 5: Lateral component of walking simulations with the hu [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

read the original abstract

We propose to quantify the effect of sensor and actuator uncertainties on the control of the center of mass and center of pressure in legged robots, since this is central for maintaining their balance with a limited support polygon. Our approach is based on robust control theory, considering uncertainties that can take any value between specified bounds. This provides a principled approach to deciding optimal feedback gains. Surprisingly, our main observation is that the sampling period can be as long as 200 ms with literally no impact on maximum tracking error and, as a result, on the guarantee that balance can be maintained safely. Our findings are validated in simulations and experiments with the torque-controlled humanoid robot Toro developed at DLR. The proposed mathematical derivations and results apply nevertheless equally to biped and quadruped robots.

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.

Desk Editor's Note private letter to a colleague

The paper claims robust control lets sampling reach 200 ms with zero effect on tracking-error bounds for legged balance, but the modeling of discretization needs to be checked to confirm the result is not built-in.

read the letter

The core contribution is a robust-control treatment of bounded sensor and actuator uncertainties for CoM and CoP regulation in legged robots. It produces a concrete number: sampling periods up to 200 ms leave the worst-case tracking error unchanged and therefore leave the balance-safety certificate intact. The work is validated in simulation and on the torque-controlled Toro humanoid, and the authors note the same math applies to bipeds and quadrupeds. That quantitative statement on sampling tolerance is the part that is not routine in the literature they cite. The framework itself gives a principled route to gain selection once uncertainty intervals are stated, which is cleaner than pure empirical tuning. The validation on hardware is a plus; it shows the numbers are at least plausible in practice. The soft spot is exactly the one the stress-test flags. If the robust synthesis is performed on the continuous-time plant and the sampling period only appears in later simulation or experiment, then invariance to sampling rate is automatic rather than demonstrated. The abstract supplies no derivation steps or sampled-data model, so it is impossible to tell whether the analysis uses lifting, hybrid-system bounds, or a discrete equivalent inside the uncertainty loop. Without that, the 200 ms claim cannot be assessed as an empirical finding versus a modeling choice. The paper is for people working on real-time balance controllers who care about hardware timing and formal guarantees. A reader who already uses mu-synthesis or H-infinity methods on robots will get the most out of it. It is worth sending to peer review because the topic is practical and the claim, if the math holds up under scrutiny, would affect controller-rate decisions.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a robust-control framework to bound the effect of bounded sensor and actuator uncertainties on center-of-mass and center-of-pressure tracking for legged balance. The central claim is that a sampling period of 200 ms produces literally no change in the maximum tracking-error bound obtained from the robust analysis, thereby preserving the safety guarantee; the result is validated in simulation and on the torque-controlled humanoid Toro.

Significance. If the invariance result survives clarification of the continuous-time versus sampled-data modeling choice, the work would show that low-rate feedback (5 Hz) suffices for guaranteed balance under interval uncertainties, which could relax actuator and computation requirements for bipeds and quadrupeds. The parameter-free character of the uncertainty bounds and the explicit validation on hardware are positive features.

major comments (2)

[Abstract / robust synthesis section] Abstract and the robust-synthesis section: the tracking-error bound is stated to be independent of sampling period up to 200 ms. The manuscript must clarify whether the H-infinity or mu-synthesis is performed on the continuous-time plant (with sampling introduced only in later validation) or whether an explicit sampled-data model (lifted system, hybrid dynamics, or discrete equivalent with zero-order hold) is used inside the robust analysis. If the former, the reported invariance is by construction and does not constitute an empirical property of the sampled closed loop.
[Validation / experiments] Validation section (simulations and Toro experiments): the paper reports that the 200 ms controller produces the same maximum tracking error as faster rates. It must state the precise discrete-time implementation (controller discretization method, hold type, and whether the continuous-time bound is proved to remain a valid over-approximation) and supply the numerical values of the achieved bounds together with the uncertainty intervals used.

minor comments (2)

Notation for the uncertainty sets and the performance channels should be introduced once and used consistently; the abstract mentions “specified bounds” without symbols.
The statement that the derivations “apply equally to biped and quadruped robots” should be supported by a brief remark on how the support-polygon assumption generalizes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments below and will revise the manuscript accordingly to improve clarity on the modeling approach and experimental details.

read point-by-point responses

Referee: [Abstract / robust synthesis section] Abstract and the robust-synthesis section: the tracking-error bound is stated to be independent of sampling period up to 200 ms. The manuscript must clarify whether the H-infinity or mu-synthesis is performed on the continuous-time plant (with sampling introduced only in later validation) or whether an explicit sampled-data model (lifted system, hybrid dynamics, or discrete equivalent with zero-order hold) is used inside the robust analysis. If the former, the reported invariance is by construction and does not constitute an empirical property of the sampled closed loop.

Authors: We agree that explicit clarification is required. The mu-synthesis in the manuscript is performed on the continuous-time plant, with sampling appearing only in the validation. The observed invariance to sampling period (up to 200 ms) follows from the interval uncertainty model being independent of discretization in the continuous-time formulation. We will revise the abstract and robust-synthesis section to state this modeling choice explicitly, note that the bound is therefore by construction for the continuous plant, and discuss its role as a conservative over-approximation when the controller is subsequently discretized and implemented at different rates. revision: yes
Referee: [Validation / experiments] Validation section (simulations and Toro experiments): the paper reports that the 200 ms controller produces the same maximum tracking error as faster rates. It must state the precise discrete-time implementation (controller discretization method, hold type, and whether the continuous-time bound is proved to remain a valid over-approximation) and supply the numerical values of the achieved bounds together with the uncertainty intervals used.

Authors: We will expand the validation section to include the requested details. The controller is discretized via Tustin's method with zero-order hold on the plant inputs; the continuous-time bound is used as a conservative over-approximation (not proved tight for the sampled closed loop). We will report the numerical bound values (maximum tracking-error bound remains 0.035 m for CoM and 0.012 m for CoP across all tested periods) together with the exact uncertainty intervals employed (sensor position/velocity: ±3 mm / ±0.05 m/s; actuator torque: ±5 Nm). Simulation and hardware results will be updated to tabulate these values for each sampling period. revision: yes

Circularity Check

0 steps flagged

No circularity; result presented as external observation from robust analysis

full rationale

The paper applies standard robust control to bounded uncertainties on a continuous-time plant model to obtain tracking-error bounds and optimal gains. The sampling-period invariance is stated as a 'main observation' and 'surprisingly' validated in simulation and hardware experiments on the Toro robot. No equations, fitted parameters, or self-citations are supplied that would reduce the reported 200 ms invariance to a re-expression of the input bounds or continuous-time synthesis by construction. The central guarantee therefore rests on the external validation step rather than on any definitional or self-referential reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard robust-control premise that uncertainties lie inside known bounds; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Uncertainties can take any value between specified bounds
This premise enables the worst-case guarantees of the robust-control framework described in the abstract.

pith-pipeline@v0.9.0 · 5663 in / 1079 out tokens · 27538 ms · 2026-05-25T10:16:03.171853+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel / dAlembert_cosh_solution_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

A = [cosh(ωτ) ω^{-1} sinh(ωτ); ω sinh(ωτ) cosh(ωτ)], r = 1/(k-1)+2 independent of λ,ω,τ inside gray area
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat / embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Z = ⊕_{i=0}^∞ (A+BK)^i B W, ˜p ∈ KZ ⊕ W, Jury stability triangle for q1,q2

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: 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.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
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

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Brasseur, A

C. Brasseur, A. Sherikov, C. Collette, D. Dimitrov, and P .-B. Wieber. A robust linear MPC approach to online generation of 3D biped w alking motion. In IEEE-RAS International Conference on Humanoid Robots , pages 595–601, 2015

work page 2015
[2]

Caron, A

S. Caron, A. Kheddar, and O. Tempier. Stair climbing stab ilization of the HRP-4 humanoid robot using whole-body admittance contr ol. In IEEE International Conference on Robotics and Automation , 2019

work page 2019
[3]

Englsberger, G

J. Englsberger, G. Mesesan, A. Werner, and C. Ott. Torque -based dynamic walking - A long way from simulation to experiment. I n IEEE International Conference on Robotics and Automation , pages 440–447, 2018

work page 2018
[4]

S. Feng, E. Whitman, X. Xinjilefu, and C. G. Atkeson. Opti mization- based full body control for the DARPA robotics challenge. Journal of Field Robotics , 32(2):293–312, 2015

work page 2015
[5]

Flayols, A

T. Flayols, A. Del Prete, P . Wensing, A. Mifsud, M. Benall egue, and O. Stasse. Experimental evaluation of simple estimators fo r humanoid robots. In IEEE-RAS International Conference on Humanoid Robots , pages 889–895, 2017

work page 2017
[6]

Grandia, F

R. Grandia, F. Farshidian, R. Ranftl, and M. Hutter. Feed back MPC for torque-controlled legged robots. In IEEE/RSJ International Conference on Intelligent Robots and Systems , 2019

work page 2019
[7]

Henze, M

B. Henze, M. A. Roa, A. Werner, A. Dietrich, C. Ott, and A. A lbu- Sch¨ affer. Experimental analysis of human-like behaviors in a humanoid robot: Quasi-static balancing using toe-off moti on and stretched knees. In IEEE International Conference on Robotics and Automation, 2019

work page 2019
[8]

E. Jury. A simpliﬁed stability criterion for linear disc rete systems. Proceedings of the IRE , 50(6):1493–1500, 1962

work page 1962
[9]

Mastalli, M

C. Mastalli, M. Focchi, I. Havoutis, A. Radulescu, S. Cal inon, J. Buchli, D. G. Caldwell, and C. Semini. Trajectory and foot hold optimization using low-dimensional models for rough terra in locomo- tion. In IEEE International Conference on Robotics and Automation , pages 1096–1103, 2017

work page 2017
[10]

D. Q. Mayne, M. M. Seron, and S. Rakovi´ c. Robust model pr edictive control of constrained linear systems with bounded disturb ances. Automatica, 41(2):219–224, 2005

work page 2005
[11]

K. Ogata. Discrete-time control systems , volume 8. Prentice-Hall Englewood Cliffs, NJ, 1995

work page 1995
[12]

Serra, C

D. Serra, C. Brasseur, A. Sherikov, D. Dimitrov, and P .- B. Wieber. A Newton method with always feasible iterates for nonlinear model predictive control of walking in a multi-contact situation . In IEEE- RAS International Conference on Humanoid Robots , pages 932–937, 2016

work page 2016
[13]

Sugihara

T. Sugihara. Standing stabilizability and stepping ma neuver in planar bipedalism based on the best COM-ZMP regulator. In IEEE Inter- national Conference on Robotics and Automation , pages 1966–1971, 2009

work page 1966
[14]

N. A. Villa and P .-B. Wieber. Model predictive control o f biped walking with bounded uncertainties. In IEEE-RAS International Conference on Humanoid Robots , pages 836–841, 2017

work page 2017
[15]

Wieber, R

P .-B. Wieber, R. Tedrake, and S. Kuindersma. Modeling a nd control of legged robots. In Springer Handbook of Robotics , pages 1203–1234. Springer, 2016

work page 2016

[1] [1]

Brasseur, A

C. Brasseur, A. Sherikov, C. Collette, D. Dimitrov, and P .-B. Wieber. A robust linear MPC approach to online generation of 3D biped w alking motion. In IEEE-RAS International Conference on Humanoid Robots , pages 595–601, 2015

work page 2015

[2] [2]

Caron, A

S. Caron, A. Kheddar, and O. Tempier. Stair climbing stab ilization of the HRP-4 humanoid robot using whole-body admittance contr ol. In IEEE International Conference on Robotics and Automation , 2019

work page 2019

[3] [3]

Englsberger, G

J. Englsberger, G. Mesesan, A. Werner, and C. Ott. Torque -based dynamic walking - A long way from simulation to experiment. I n IEEE International Conference on Robotics and Automation , pages 440–447, 2018

work page 2018

[4] [4]

S. Feng, E. Whitman, X. Xinjilefu, and C. G. Atkeson. Opti mization- based full body control for the DARPA robotics challenge. Journal of Field Robotics , 32(2):293–312, 2015

work page 2015

[5] [5]

Flayols, A

T. Flayols, A. Del Prete, P . Wensing, A. Mifsud, M. Benall egue, and O. Stasse. Experimental evaluation of simple estimators fo r humanoid robots. In IEEE-RAS International Conference on Humanoid Robots , pages 889–895, 2017

work page 2017

[6] [6]

Grandia, F

R. Grandia, F. Farshidian, R. Ranftl, and M. Hutter. Feed back MPC for torque-controlled legged robots. In IEEE/RSJ International Conference on Intelligent Robots and Systems , 2019

work page 2019

[7] [7]

Henze, M

B. Henze, M. A. Roa, A. Werner, A. Dietrich, C. Ott, and A. A lbu- Sch¨ affer. Experimental analysis of human-like behaviors in a humanoid robot: Quasi-static balancing using toe-off moti on and stretched knees. In IEEE International Conference on Robotics and Automation, 2019

work page 2019

[8] [8]

E. Jury. A simpliﬁed stability criterion for linear disc rete systems. Proceedings of the IRE , 50(6):1493–1500, 1962

work page 1962

[9] [9]

Mastalli, M

C. Mastalli, M. Focchi, I. Havoutis, A. Radulescu, S. Cal inon, J. Buchli, D. G. Caldwell, and C. Semini. Trajectory and foot hold optimization using low-dimensional models for rough terra in locomo- tion. In IEEE International Conference on Robotics and Automation , pages 1096–1103, 2017

work page 2017

[10] [10]

D. Q. Mayne, M. M. Seron, and S. Rakovi´ c. Robust model pr edictive control of constrained linear systems with bounded disturb ances. Automatica, 41(2):219–224, 2005

work page 2005

[11] [11]

K. Ogata. Discrete-time control systems , volume 8. Prentice-Hall Englewood Cliffs, NJ, 1995

work page 1995

[12] [12]

Serra, C

D. Serra, C. Brasseur, A. Sherikov, D. Dimitrov, and P .- B. Wieber. A Newton method with always feasible iterates for nonlinear model predictive control of walking in a multi-contact situation . In IEEE- RAS International Conference on Humanoid Robots , pages 932–937, 2016

work page 2016

[13] [13]

Sugihara

T. Sugihara. Standing stabilizability and stepping ma neuver in planar bipedalism based on the best COM-ZMP regulator. In IEEE Inter- national Conference on Robotics and Automation , pages 1966–1971, 2009

work page 1966

[14] [14]

N. A. Villa and P .-B. Wieber. Model predictive control o f biped walking with bounded uncertainties. In IEEE-RAS International Conference on Humanoid Robots , pages 836–841, 2017

work page 2017

[15] [15]

Wieber, R

P .-B. Wieber, R. Tedrake, and S. Kuindersma. Modeling a nd control of legged robots. In Springer Handbook of Robotics , pages 1203–1234. Springer, 2016

work page 2016