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Optimization free control and ground force estimation with momentum observer for a multimodal legged aerial robot

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Pith's one-line read An optimization-free reference filter prevents foot slip on slippery ground, and a momentum observer recovers ground reaction forces in simulation.

desk verdict A plausible simulation study combining an established ERG with a momentum observer for legged-aerial robots, but the friction constraint as written does not enforce |Fy| ≤ μFz and the evidence is a single 10 s run. read the letter →

arxiv 2411.11216 v1 pith:QKNPSC5J submitted 2024-11-18 cs.RO cs.SYeess.SY

classification cs.ROcs.SYeess.SY
keywords legged-aerialrobotexplicitreferencegovernormomentumobservergroundreactionforceestimationfrictionconeconstraintsreduced-ordermodelthruster-assistedwalkingHusky
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a lightweight walking controller for a legged-aerial robot that replaces online optimization with an Explicit Reference Governor (ERG): a filter that adjusts the commanded body velocity until predicted ground reaction forces stay inside a friction pyramid, so the feet do not slip even on a low-friction surface (coefficient 0.25 in simulation). The same framework feeds thruster forces from an attitude controller into the governor, letting thrust assist walking without a heavy onboard solver. The paper also tests a conjugate momentum observer that estimates the total ground reaction wrench from proprioceptive states and known thruster inputs, and shows it tracks the simulated ground forces better than a constrained-model estimator, which loses rank during two-point contact. The motivation is that legged-aerial robots have conflicting payload and computing requirements, so an optimization-free controller that still respects friction constraints could make low-power onboard computers viable.

What carries the argument

The two load-bearing mechanisms are the Explicit Reference Governor and the conjugate momentum observer. The ERG is a filter on the commanded reference that enforces constraints by projecting the reference onto the constraint-admissible set; here the constraint $h_r$ is a friction pyramid built from predicted ground reaction forces in the triangular inverted pendulum model, so satisfying $h_r \geq 0$ means the foot force lies inside the friction cone and should not slip. The momentum observer defines a residual $r = K_O \bigl(p(t) - \int_0^t (r - \hat\beta + B_t u_t)\,dt\bigr)$ whose steady state approaches the external ground wrench as the observer gain grows, avoiding joint-acceleration measurements and mass-matrix inversion. The constrained-model estimator, by contrast, pseudo-inverts $J M^{-1} J^\top$, and that matrix loses rank during two-point contact, which the paper identifies as the reason it fails to track the impulsive ground forces.

What would settle it

Run the ERG controller on the physical Husky on a floor with friction coefficient 0.25 while recording stance-foot velocity; if any stance foot's horizontal velocity relative to the ground exceeds the slip threshold defined by the Stribeck model while the computed constraint $h_r \geq 0$ is satisfied, the reduced-order friction-cone guarantee is falsified.

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Extended reading notes

Core claim

On the authors' own terms, the central claim is that foot slip can be prevented without solving any optimization problem at run time. The ERG treats the robot as a triangular inverted pendulum in two-point contact, computes a friction-pyramid constraint $h_r \geq 0$ from the predicted ground reaction forces, and filters the reference velocity so the applied gait never leaves the admissible set. Simulated on the Husky reduced-order model with a Coulomb/Stribeck friction coefficient of 0.25, the controller keeps the feet from slipping and also places feet laterally to manage roll caused by thruster forces. The second claim is that a conjugate momentum observer with sufficiently high gains recovers the sum total of ground reaction forces from the generalized momentum residual, and does so more faithfully than a constrained ground model, whose Delassus matrix $J M^{-1} J^\top$ is rank-deficient during two-point contact.

Load-bearing premise

The no-slip guarantee rests on a reduced-order model in which the body is a point mass with massless legs, the gait is a perfect two-point contact, the lateral ground forces are assumed evenly distributed, and the thrusters act only upward in the body frame; if any of these fail on the real robot, foot slip can occur despite the governor's constraint being satisfied.

Editorial extensions

If this is right

  • Walking control can run on low-power real-time hardware because it requires only reference filtering and constraint evaluation, not repeated convex optimization.
  • The same ERG formulation extends naturally to thrust-assisted gaits: the attitude controller's thruster wrench enters the force balance the governor uses to predict ground reactions.
  • Ground reaction forces can be estimated without joint accelerometers by integrating residual momentum, which is useful for contact and terrain monitoring on multimodal robots.
  • In two-point contact, constrained-model force estimation is structurally unreliable because the Delassus matrix is rank-deficient, so momentum-based estimation is the better fit for this gait.
  • With friction coefficient 0.25, the filtered reference selects foot placements, including lateral placement, that keep contact forces inside the friction pyramid.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's simulation assumes massless legs and thrusters that act only upward in the body frame; if those assumptions fail on hardware, the actual contact force could leave the friction cone even though the computed constraint $h_r \geq 0$ is satisfied, so slip could reappear.
  • The same residual signal used for ground force estimation could serve as a contact detector or terrain classifier, since its impulsive peaks coincide with foot-strike events in the simulation.
  • Adding leg inertia or joint compliance to the reduced-order simulation is a direct test of how much of the no-slip guarantee depends on the massless-leg assumption.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes an optimization-free control and estimation framework for the Husky legged-aerial robot in a reduced-order simulation. A modified Explicit Reference Governor (ERG) filters velocity references to keep predicted ground reaction forces inside a friction pyramid, with the goal of preventing foot slip on slippery ground (mu=0.25), while also incorporating thruster forces from an attitude controller. A conjugate momentum observer is used to estimate the total ground reaction force wrench, and its output is compared against a constrained-model estimator. The claims are supported by a 10-second simulation of a two-point-contact gait, with qualitative plots of body states, foot states, constraint satisfaction, and force estimates.

Significance. If the results hold, the paper offers a useful step toward real-time legged-aerial control: it replaces online optimization for friction-cone enforcement with an explicit governor and applies a standard momentum observer for ground reaction force estimation. The momentum observer equations in Section IV-A are standard and correctly avoid joint-acceleration measurement and mass-inertia inversion. The simulation is not circular in the obvious sense: the estimator is not fed the target ground reaction forces, and the observer gain is not fitted to the ground model. The paper also candidly acknowledges the limitations of the reduced-order model and the need for hardware validation. However, the central no-slip guarantee rests on Eq. (10), which as written does not enforce a two-sided lateral friction cone, and the quantitative evidence in Section V is too thin to fully support the claimed tracking and anti-slip performance.

major comments (3)
  1. [Section III-A, Eq. (10)] The second row of the friction constraint is [0, -sgn(u_{gi,x}), mu_s], which enforces mu_s F_z >= sgn(F_x) F_y, not mu_s F_z >= |F_y|. As written, the admissible set is not the friction cone, so a reference state with |F_y| > mu_s F_z can satisfy the constraint whenever the sign of F_y opposes the sign of F_x. The correct row should use -sgn(u_{gi,y}) in the second entry. Fig. 8 plots only F_x against mu*F_z and never checks F_y, so the evidence in the paper does not close this gap. If the simulation code contains the correct sign, Eq. (10) is a misprint that must be corrected; if the code follows Eq. (10), the 'no slippage at the foot end' claim is unsupported for lateral forces. In addition, the vector d_r in Eq. (10) appears to contain four entries (0, 0, -u_min, g_{i,z}) while the matrix J_r is 3x3, so the dimensions are inconsistent; this must also be fixed.
  2. [Section III-A] The paper states that the ERG 'incorporates external thruster forces from an attitude controller,' but it never shows the ERG update law or the map from the filtered velocity reference and the applied thruster wrench u_t to the predicted ground reaction forces u_{g,i}. Eq. (9) only states the translational dynamics, and Eq. (10) only defines the constraint set; the construction of A and b, and the way u_t enters the constraint prediction, are omitted and referred to [18]. Since the thruster-aware modification is a stated contribution, this derivation should not be outsourced to a reference; the authors should include the explicit update rule or a self-contained derivation so that the claim can be checked.
  3. [Section V] The validation consists of one 10-second simulation run with qualitative plots. The claims that the foot 'doesn't slip' and that the momentum observer 'closely tracks' ground reaction forces are not accompanied by quantitative metrics: there is no foot slip distance or slip margin over the run, no RMSE or peak error for the F_x/F_y/F_z estimates, no convergence time for the observer, and no comparison against a baseline without the ERG or with different observer gains. The observer gain K_O = 1000*I_6 is stated but no sensitivity analysis is reported. The results section should include such metrics to substantiate the central claims.
minor comments (4)
  1. [Throughout] The manuscript contains several typos and grammatical errors, including 'V orndamme' in the references, 'the the' in the introduction, and 'the ERG is able to efficiently able to find' in Section V; a careful proofread is needed.
  2. [Section II, after Eq. (5)] The sentence 'The term ut ∈ R6 ut represents the actions exerted by the four thrusters' has a doubled 'ut'; it should read 'The term ut ∈ R6 represents ...'.
  3. [Figures 8 and 10] The y-axis labels in Fig. 8 do not state units, and Fig. 10 has no in-plot legend distinguishing the actual, estimated, and constrained-model traces; adding units and legends would improve readability.
  4. [Section V] The statement 'The momentum observer is able to observe the sum total of the ground reaction forces shown in Fig. 10' should be rephrased to clarify that this is a simulation result for the reduced-order model, not a hardware observation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the ERG and momentum-observer results are validated against the simulation dynamics rather than forced by fitted inputs or self-referential definitions.

full rationale

The paper's central claims are that an optimization-free ERG can filter velocity references so that planned ground reaction forces respect a friction pyramid, and that a conjugate momentum observer can estimate the summed ground reaction forces in a reduced-order Husky simulation. Neither claim reduces to its inputs by construction. The ERG constraint in Eq. (10) is derived from the point-mass, two-point-contact model of Eq. (9), while the actual contact forces in simulation are produced by the Stribeck ground model of Eq. (8); the ERG does not inject its constraint into the ground model, so the no-slip result exhibited in Fig. 8 is a genuine closed-loop outcome rather than a direct assignment. The momentum observer in Section IV-A uses the standard residual dynamics r = K_O(p - ∫(r - β + B_t u_t)dt) with a fixed gain K_O = 1000 I_6; the gain is not fitted to the target forces, and the agreement shown in Fig. 10 is an independent comparison against the spring-damper ground model. The dependence on the authors' earlier ERG work [18] for the reference-update law is a normal citation of an algorithmic formulation, not an import of this paper's conclusion, and the present simulation provides the supporting evidence for the claimed behavior. A potential correctness concern about Eq. (10) (the second row uses -sgn(u_x) rather than -sgn(u_y), so the lateral friction cone may not be enforced as written) is a technical error, not circularity, and therefore does not raise the circularity score.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claims rest on several simplifying modeling choices: massless legs, a triangular inverted pendulum ERG model, thrusters acting only upward, and a homogeneous ground friction model. The observer also assumes an exact dynamic model and accurate thruster characterization. No new physical entities are introduced; the free parameters are observer and controller gains plus ground model coefficients, most of which are unreported.

free parameters (4)
  • Observer gain K_O = 1000*I6
    Chosen for the simulation; sets convergence rate of momentum residual to GRF estimate; not tuned systematically (Fig. 10 caption).
  • Static friction coefficient mu_s = 0.25
    Used in the ERG friction-cone constraint and in the Stribeck ground model to represent slippery ground (Section V).
  • Attitude controller gains Kp, Kd
    Used to generate thruster wrench in Eq. (11); values not reported in the paper.
  • Ground model parameters kgz, kdz, mu_c, mu_v, vs
    Shape the spring-damper and Stribeck friction ground response in Eq. (8); values not specified, so the simulated 'actual' forces are not reproducible.
assumptions (6)
  • domain assumption Legs are massless and the body is a point mass with inertia, so leg states are ignored in the dynamics.
    Section II: 'The legs of HROM are massless, so we can ignore all leg states.'
  • domain assumption Thrusters generate force only upward in the body frame.
    Section II: 'The individual thruster forces are modeled as forces that can act only upwards in the body frame.'
  • domain assumption Ground contact forces are modeled by a compliant spring-damper ground with Stribeck friction.
    Section II Eq. (8) defines the ground reaction force model used as 'actual' in simulation.
  • domain assumption The ERG constraint is derived from a relaxed triangular inverted pendulum with two-point contact and assumes even lateral force distribution.
    Section III-A: 'It assumes the robot to be a triangular inverted pendulum with a point mass... It can be assumed that the lateral ground forces are distributed evenly.'
  • domain assumption Thruster PWM input maps one-to-one to known thrust under stable nominal voltage.
    Section IV-A: 'we also assume that the thrusters have been accurately characterized and there is a one to one mapping to the PWM input and the thrust.'
  • standard math Momentum observer converges to external wrench as observer gain tends to infinity (result from Haddadin et al.).
    Section IV-A, Eq. (12), citing [2]. This is a standard result in momentum observer theory.

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Cite this review

Pith. "Pith review of Optimization free control and ground force estimation with momentum observer for a multimodal legged aerial robot." pith.science (2026). https://pith.science/paper/QKNPSC5J

@misc{pith2026241111216,
  author       = {Pith},
  title        = {Pith review of: Optimization free control and ground force estimation with momentum observer for a multimodal legged aerial robot},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QKNPSC5J}},
  note         = {Machine review of arXiv:2411.11216}
}
read the original abstract

Legged-aerial multimodal robots can make the most of both legged and aerial systems. In this paper, we propose a control framework that bypasses heavy onboard computers by using an optimization-free Explicit Reference Governor that incorporates external thruster forces from an attitude controller. Ground reaction forces are maintained within friction cone constraints using costly optimization solvers, but the ERG framework filters applied velocity references that ensure no slippage at the foot end. We also propose a Conjugate momentum observer, that is widely used in Disturbance Observation to estimate ground reaction forces and compare its efficacy against a constrained model in estimating ground reaction forces in a reduced-order simulation of Husky.

Figures

Figures reproduced from arXiv: 2411.11216 by the authors.

Figure 1
Figure 1. Husky fitted with the Electric Ducted fans. Each leg is actuated [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Depiction of the robot and HROM parameters. [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Flowchart of Control and estimation framework. The momentum [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Explicit Reference Governor (ERG) manipulates the applied [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Model used in the ERG controller to estimate the ground reaction [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]
Figure 8
Figure 8. Figure 8: Plots of constraint satisfaction with mu = 0.25. [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 9
Figure 9. Figure 9: Plots of leg angle and lengths during simulations [PITH_FULL_IMAGE:figures/full_fig_p005_9.png]
Figure 10
Figure 10. Figure 10: Estimated Ground reaction forces from the momentum estimator [PITH_FULL_IMAGE:figures/full_fig_p006_10.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.