Conjugate momentum based thruster force estimate in dynamic multimodal robot

Adarsh Salagame; Alireza Ramezani; Chenghao Wang; Eric Sihite; Kaushik Venkatesh Krishnamurthy; Morteza Gharib; Reza Nemovi; Shreyansh Pitroda; Taoran Liu

REVIEW 4 major objections 6 minor 2 cited by

Conjugate momentum based thruster force estimate in dynamic multimodal robot

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A conjugate-momentum observer estimates thruster forces on a legged-aerial biped to within 5–19 percent error.

desk verdict Standard momentum observer, honestly applied to a new platform; simulation-only and no sensitivity analysis, but a plausible niche contribution. read the letter →

arxiv 2411.14596 v1 pith:VKT33ZQZ submitted 2024-11-21 cs.RO

Shreyansh Pitroda , Eric Sihite , Taoran Liu , Kaushik Venkatesh Krishnamurthy , Chenghao Wang , Adarsh Salagame , Reza Nemovi , Alireza Ramezani

show 1 more author

Morteza Gharib

This is my paper · ORCID

classification cs.RO

keywords conjugatemomentumobserverthrusterforceestimationlegged-aeriallocomotionbipedalrobotgroundreactioncontactconstraintmodelHarpyexternalwrench

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 conjugate-momentum observer that estimates thruster forces online for Harpy, a bipedal robot with electric ducted fans that can walk and fly. The aim is to establish that the observer can track the true generalized thruster forces while the robot performs thruster-assisted walking in simulation, avoiding the need for thrust-stand calibration that misses working conditions such as battery voltage drop. Estimation requires ground-reaction-force information; the paper shows two routes: feeding the compliant ground model's forces directly, or estimating them from a contact-constraint model. With the constraint model, normalized root-mean-square errors for generalized forces and torques lie between 0.049 and 0.195. The authors argue this is accurate enough to feed controllers such as MPC and QP, and that the constraint-model route removes the need for foot force sensors.

What carries the argument

The central object is the conjugate momentum observer, a standard momentum observer built on the generalized momentum $p = M\dot q$. The estimated thruster force $r$ evolves as $\dot r = K_0(B_t u_t - r)$, making $r$ a low-pass filter of the true thruster force with gain $K_0$; no joint accelerations or inertia-matrix inversions are needed. Because the observer equation needs ground reaction forces, the paper pairs it with a contact-constraint model that enforces zero stance-foot acceleration, $J_c \ddot q = \dot J_c \dot q$, to compute the ground reaction force via a Moore-Penrose pseudo-inverse. The controller used for the simulated walking is built on a variable-length inverted pendulum (VLIP) reduced-order model, with thrusters stabilizing roll and yaw, while the full simulation uses an Euler-Lagrangian model with compliant ground contact.

What would settle it

Measure the actual thruster force with a thrust stand while Harpy walks with thrusters on hardware and compare it with the observer's output: if the normalized RMSE grows well beyond the 0.05–0.19 range under realistic model mismatch, joint friction, or foot slip, the central claim fails. Alternatively, instrument the foot with a force plate to check whether the stance-foot acceleration is truly zero while the constraint model is active.

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Extended reading notes

Core claim

Using the generalized momentum $p = M\dot q$, the observer defines an estimated generalized thruster force $r$ driven by $\dot r = K_0(B_t u_t - r)$, so that $r$ is a low-pass filtered version of the actual thruster force. Under ideal conditions, $M$ and $\beta$ are assumed exactly known. The body-frame thruster force is recovered with the Jacobian pseudo-inverse $\hat u_t = [J_t^\top J_t]^\dagger J_t^\top r$. When ground reaction forces are taken from the compliant ground model, the estimated generalized forces and torques match the actual values closely. When they are instead estimated through the contact constraint $J_c \ddot q = \dot J_c \dot q$, the estimates still track but with larger error, and the paper reports NRMSE values of 0.1156 for $F_x$, 0.0492 for $F_y$, 0.1946 for $F_z$, 0.1102 for $\tau_x$, 0.1342 for $\tau_y$, and 0.1239 for $\tau_z$. The vertical force $F_z$ has the largest error, which the authors attribute to the constraint model.

Load-bearing premise

The estimator assumes the robot's mass and bias terms are known exactly, and that the stance foot never accelerates or slips while the contact constraint is enforced; onboard hardware will violate both assumptions to some degree.

Editorial extensions

If this is right

Controllers such as MPC and QP can use the estimated thruster forces directly, replacing thrust-stand calibration that misses battery voltage and other working conditions.
A contact-constraint-based ground reaction force estimate removes the need for foot force sensors, at the cost of higher estimation error, with $F_z$ the least accurate.
Because the estimator is a low-pass filter, its accuracy is governed by the quality of the ground reaction force information; better terrain knowledge yields better thrust estimates.
The double-support phase makes $J_c M^{-1} J_c^\top$ rank-deficient, so ground force estimates are inherently less reliable during that phase.
A second-order filter is the paper's stated next step to reduce the filter lag visible in the y-direction torque estimate.

Reading between the lines

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

On hardware, the ideal-model assumption will be violated by joint friction and model error; the paper gives no sensitivity analysis, so the first test is whether the observer gain can absorb these errors without amplifying noise.
The rank deficiency in double support suggests a practical deployment would gate the estimator by contact state or blend the constraint-model ground reaction force with any available foot force sensing.
The same observer structure could be transferred to other thruster-augmented legged robots, since it only requires the generalized momentum, contact Jacobians, and a source of ground reaction force; the equations do not depend on Harpy's specific kinematics.
A direct experimental check would be to compare the observer's thrust estimate against a thrust stand while battery voltage drops during flight, quantifying how much of the 5–19% error persists outside simulation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Quick take: this is a textbook momentum observer applied to Harpy, the thruster-assisted biped, in simulation. What is new is not the estimator but the platform and the specific comparison: feeding the observer with terrain information from a contact constraint model versus a known ground model, and showing the constraint-model version still tracks thruster forces with NRMSE 0.05-0.19. That is a modest but real step for the legged-aerial subfield.

The paper does a few things well. The derivation is standard but clean, and the authors are honest about the limitations: they explicitly say the constraint-model estimate is coupled to the residual it feeds, and they note that any error in the observer will propagate into the GRF estimate. They also flag the double-support rank deficiency. That level of candor is welcome.

The soft spots are proportional to the contribution. It is simulation-only, with no code or data release, so the numbers in Table I are not independently checkable. The observer gains are hand-tuned—K0 jumps from 25 to 3000 between the ground-model and constraint-model cases—and there is no sensitivity analysis for model error, friction, or foot slip. The paper assumes M-hat = M and beta-hat = beta, which is fine for an ideal simulation but not for hardware. The sign typo in the constraint equation (Jc qddot = Jcdot qdot instead of the negative) is minor and does not appear to carry into equation (16), which has the right sign. Also, the comparison to the iRonCub EKF is too brief; a direct baseline would have made the claim stronger.

The stress-test note about the coupled loop is correct, but the paper already concedes it. That does not make the weakness go away—it remains unquantified—but it is not a hidden flaw. The central claim, that the observer can track thruster forces under exact-model simulation, holds up. What is not demonstrated is robustness under the conditions Harpy will actually see.

Who is this for? People working on legged-aerial robots, especially the Harpy project itself. A serious referee should be able to ask for code, a sensitivity analysis, and ideally a hardware trial. I would accept it for peer review with revisions, though I would not cite it myself unless I worked on that platform.

Recommendation: send it out, with the understanding that the authors need to release code/data and address the robustness gap.

Referee Report

4 major / 6 minor

Summary. The paper proposes a generalized-momentum (conjugate momentum) observer to estimate thruster forces on the Harpy bipedal robot, a platform that combines legged locomotion with thruster actuation. The observer is derived in Section III and is evaluated in simulation during thruster-assisted walking. Two variants are presented: one that uses ground reaction forces from the compliant ground model directly, and one that estimates ground reaction forces from a contact constraint model. The reported results are normalized RMSE values between 0.05 and 0.19 for the generalized force/torque components in the constraint-model case (Table I). The paper claims that the observer accurately tracks thruster forces and that the method is suitable for onboard estimation where thrust-stand characterization is insufficient.

Significance. If the result holds, the paper offers a sensorless thruster-force estimation scheme for a multimodal legged-aerial robot, which is relevant for downstream MPC/QP control. A strength is that the paper builds on the well-established momentum-observer framework, avoiding acceleration estimation and matrix inversion. Another strength is that the simulation includes full-dynamics evaluation and reports quantitative errors, not just qualitative plots. However, the contribution is incremental relative to the existing momentum-observer literature, and the reported accuracy is obtained under strong ideal-model assumptions. The paper does not yet demonstrate robustness to model error, foot slip, or state-estimation delay, which are central to the stated motivation of onboard estimation under real operating conditions.

major comments (4)

[III, Eq. (13)] The observer equation is not self-contained: the equality rdot = K0(Bt ut - r) = K0(pdot - pdot_hat) introduces pdot_hat without a definition or an explicit relation to the state and known inputs. In the standard momentum-observer derivation, one obtains rdot = K0(pdot - beta - r - Bg lambda - Bj uj), and the equivalence to K0(pdot - pdot_hat) must be shown. Please define pdot_hat and state the exact residual dynamics used in the simulator.
[III-A] The zero-foot-acceleration constraint is stated as Jc qddot = Jdot qdot, but differentiating Jc qdot = 0 gives Jc qddot = -Jdot qdot. Equation (16) contains the correct minus sign, so this is an inconsistency in the text rather than in the implemented formula; please correct the sentence to avoid confusion.
[IV-B] The constraint-model estimator is a feedback loop: lambda in Eq. (16) is computed from the residual r, and r is integrated in Eq. (14) using that lambda. No passivity, small-gain, or error-propagation analysis is given for this loop, and the only mitigation offered is raising K0 from 25 to 3000. Because Table I is generated under the ideal assumptions M-hat = M, beta-hat = beta, and no foot slip, the reported 0.05-0.19 NRMSE is not yet demonstrated as robust to the model error and terrain variation expected on hardware. Please add a sensitivity analysis (e.g., parameter perturbations, friction/slip, state-estimation delay) or a bounded-error argument.
[IV, Table I] The stated contribution is a comparison of estimation with and without terrain knowledge, but Table I reports NRMSE only for the constraint-model case. The ground-model case (Fig. 6) is only qualitative. Please report the same NRMSE metric for both cases so the comparison is quantitative.

minor comments (6)

[Abstract and Section I] There are numerous grammatical errors and typos, such as 'such our state-of-the-art Harpy platform' and 'we can characterize thruster force using a thrust stand but it generally does not account for working conditions.' Please proofread the manuscript.
[II] Harpy's height is stated as 600 cm, which is presumably a typo for 60 cm. Please correct the unit.
[II-C, Eq. (9)] The composition of u_t from u_t,c and u_t,L/u_t,R is unclear; specify the dimensions and the frames in which the components are expressed.
[IV-B] The observer gains are reported for the two cases, but no tuning procedure or criterion is given. State how the gains were selected and whether the results are sensitive to their values.
[IV-B, Table I] The 'Normalized RMSE' metric is not defined. Specify the normalization denominator (e.g., range, mean, standard deviation) so that the values in Table I are interpretable.
[III] The term 'conjugate momentum' is used, but the derivation is the standard generalized momentum observer. Consider clarifying the terminology or providing a reference that uses this name.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the thruster-force estimator is a standard momentum-based observer, and its validation against simulated ground truth does not reduce to its own inputs by construction.

full rationale

The estimation scheme is the classical generalized-momentum observer: equation (13) defines the residual dynamics as dot-r = K0 (Bt ut - r) under the stated ideal-model assumption M-hat = M and beta-hat = beta, so the estimated thruster force is a low-pass filtered version of the actual thruster force rather than an algebraic renaming of it. The observer gains are hand-tuned and are not fitted to the NRMSE values reported in Table I, and the reported errors are comparisons against simulated ground truth, not a fitted parameter renamed as a prediction. The constraint-model branch (equation (16)) does feed the estimated residual r back into the GRF estimate, and that GRF estimate is then used in equation (14); this creates a coupled observer loop, but it is a genuine feedback loop with error dynamics, not a definitional identity, and the paper explicitly acknowledges that errors in r propagate into the GRF estimate and that the observer gain must be increased to mitigate this. The remaining limitations (exact model knowledge, zero stance-foot acceleration, no slip) are stated assumptions and affect external validity, but they are not circular reductions. Self-citations to prior Harpy hardware and control work are contextual and do not carry the load-bearing derivation, which rests on the standard momentum-observer literature and the paper's own dynamics equations.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The estimator itself is a textbook momentum observer; the load-bearing additions are the assumptions of exact model knowledge and a no-slip contact constraint. The only numbers fitted to make the results look good are the observer gains. No new physical entities are introduced.

free parameters (3)

Observer gain K0 = K0 = diag(1,1,1,1,25,25,25,25,25,25) for ground-model case; K0 = diag(1,1,1,1,800,1200,60,3000,800,500) for…
Hand-picked per scenario; no tuning rule or sensitivity analysis, and the constraint-model case uses much higher gains to overcome the coupled estimation loop.
Ground contact model parameters = µs=0.8, µc=0.64, µv=0.8, kg,p=8000, kg,d=268
These define the simulated ground response used as ground truth in one validation case; the estimator's reported accuracy depends on this particular contact model.
Robot mass, inertia, and link dimensions = mB=2, mH=mK=0.5, IB=1e-3, IH=IK=1e-4; l1=[0,0.1,-0.1], l2=[0,0.5,0], l3=[0,0,-0.3], l4=[0,0.1,0]
Used in both the plant and the observer; the estimator assumes exact knowledge, so the paper does not test robustness to parameter error.

assumptions (3)

domain assumption The observer dynamic model is exact: M-hat = M and beta-hat = beta.
Stated in Section III: 'Under ideal conditions, we assume M-hat = M and beta-hat = beta'. If the plant differs from the model (friction, payload, battery voltage), the estimated thruster force has no guaranteed accuracy.
domain assumption The stance foot acceleration is zero and there is no slip during contact.
Used in Section III-A to derive the contact constraint GRF estimate (16). The no-slip assumption also appears in the VLIP model in Section II-B. Slip or soft terrain would invalidate the constraint.
domain assumption Thrusters can produce force in any direction and their dynamics are ignored.
Section II states 'we assume that the thrusters can provide forces in any direction, and the thruster dynamics are ignored.' This simplifies the estimator but is not tested against actuator lag or saturation.

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Cite this review

Pith. "Pith review of Conjugate momentum based thruster force estimate in dynamic multimodal robot." pith.science (2026). https://pith.science/paper/VKT33ZQZ

@misc{pith2026241114596,
  author       = {Pith},
  title        = {Pith review of: Conjugate momentum based thruster force estimate in dynamic multimodal robot},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VKT33ZQZ}},
  note         = {Machine review of arXiv:2411.14596}
}

read the original abstract

In a multi-modal system which combines thruster and legged locomotion such our state-of-the-art Harpy platform to perform dynamic locomotion. Therefore, it is very important to have a proper estimate of Thruster force. Harpy is a bipedal robot capable of legged-aerial locomotion using its legs and thrusters attached to its main frame. we can characterize thruster force using a thrust stand but it generally does not account for working conditions such as battery voltage. In this study, we present a momentum-based thruster force estimator. One of the key information required to estimate is terrain information. we show estimation results with and without terrain knowledge. In this work, we derive a conjugate momentum thruster force estimator and implement it on a numerical simulator that uses thruster force to perform thruster-assisted walking.

Figures

Figures reproduced from arXiv: 2411.14596 by the authors.

**Figure 2.** Illustrates Reduced-order, variable length model with a point-mass [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** An overview of the estimator algorithm. where, ˆh = Gˆ(qˆ) +Cˆ [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Illustrates estimated generalized thruster force from Conjugate [PITH_FULL_IMAGE:figures/full_fig_p004_5.png]

**Figure 6.** Figure 6: Comparison of GRF from ground model and Constraint model. [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]

**Figure 8.** Figure 8: Illustrates the simulated robot states. REFERENCES [1] S. Haddadin, A. De Luca, and A. Albu-Schaffer, “Robot Collisions: ¨ A Survey on Detection, Isolation, and Identification,” IEEE Transactions on Robotics, vol. 33, no. 6, pp. 1292–1312, Dec. 2017. [2] A. de Luca an…

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

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Reference graph

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Pith tools

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

[1] [1]

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S. Haddadin, A. De Luca, and A. Albu-Sch ¨affer, “Robot Collisions: A Survey on Detection, Isolation, and Identification,” IEEE Trans- actions on Robotics , vol. 33, no. 6, pp. 1292–1312, Dec. 2017

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A. De Luca and R. Mattone, “Actuator failure detection and isolation using generalized momenta,” in 2003 IEEE International Conference on Robotics and Automation (Cat. No.03CH37422) , vol. 1, Sep. 2003, 634–639 vol.1

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G. Garofalo, N. Mansfeld, J. Jankowski, and C. Ott, “Sliding Mode Momentum Observers for Estimation of External Torques and Joint Acceleration,” in 2019 International Conference on Robotics and Automation (ICRA), May 2019, pp. 6117–6123

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