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REVIEW 3 major objections 8 minor 33 references

Koopman Representation of Nonlinear Virtual Environments in Kinesthetic Haptic Systems

T0 review · 3 major / 8 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that the nonlinear dynamics of a virtual environment can be lifted into a linear Koopman model that supports real-time haptic rendering and a closed-loop stability analysis that is less conservative than passivity-based…

desk verdict Koopman/EDMD is applied cleanly to haptic rendering and the experiments are credible, but the headline stability guarantee is computed on a purely linear recurrence that does not match the relifted implementation; fix that gap and the paper is a solid applied contribution. read the letter →

arxiv 2608.11461 v1 pith:5M7S4PYT submitted 2026-08-11 cs.RO

Yanting Zhou , Jozsef Kövecses , James Richard Forbes This is my paper · ORCID
classification cs.RO
keywords Koopmanoperatorhapticrenderingnonlinearvirtualenvironmentclosed-loopstabilityDuffingoscillatorpassivitydata-drivenmodeleigenvalueanalysis
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 claims that a data-driven Koopman representation of a nonlinear virtual environment can serve as the model used for real-time haptic rendering, and that because the representation is linear in a lifted state space, the stability of the whole device–coupling–environment loop can be certified just by checking the eigenvalues of a closed-loop matrix. The support is a Duffing-oscillator virtual environment: a Koopman model trained from simulated data reproduces the nonlinear model's responses in simulation and experiment, and a multi-user study indicates users cannot reliably distinguish the two. The authors further claim this eigenvalue-based stability test is less conservative than passivity-based sufficient conditions, and that the Koopman rendering is more robust to errors in the assumed rigid-body device model than the original nonlinear model. If true, the practical payoff is a design workflow in which virtual coupling gains and sampling parameters can be chosen directly from a linear stability condition without sacrificing the fidelity of nonlinear force feedback.

What carries the argument

The load-bearing object is the Koopman representation of the virtual environment: a set of lifting functions $\vartheta(x_k)$ that embed the VE state into a higher-dimensional space in which the nonlinear update is approximated as linear, $\vartheta_{k+1} = A\vartheta_k + B u_k$ with residual $r_k$. The lifted state vector here uses third-order monomials of position and velocity, and the matrices $A$ and $B$ are fit by least squares from simulated trajectories of the Duffing oscillator. This linear model is then coupled to the discrete-time device model through the virtual-coupling force, producing the closed-loop matrix $A_{\mathrm{CL}}$ in Eq. (19); stability is decided by whether all eigenvalues of $A_{\mathrm{CL}}$ lie strictly inside the unit circle. The same linear-in-the-lifted-space structure is what the authors use to explain robustness to device-modeling uncertainty: perturbations in the VE position enter through the linear combination of lifted states rather than being amplified by a standalone cubic term.

What would settle it

Run the closed-loop haptic simulation with the same parameters used in the paper but with added noise or drift in the lifted states, and compare the trajectories with the eigenvalue prediction; if a trajectory diverges while all eigenvalues of the closed-loop matrix lie inside the unit circle, the stability claim does not cover the implemented relifting loop. Alternatively, measure the per-step residual of the Koopman model on held-out trajectories; if the residual is large enough to move an eigenvalue across the unit circle, the certificate is void.

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Extended reading notes

Core claim

The core discovery is that the nonlinear dynamics of a virtual environment need not be simulated with a nonlinear model or stabilized with passivity arguments; they can be lifted into a higher-dimensional linear state-space model via the Koopman operator, and that lifted model can be inserted directly into the closed-loop haptic system. For the 1-DoF Duffing-oscillator VE, lifting functions up to third-order monomials in position and velocity produce a Koopman model whose simulated and experimental responses match the nonlinear baseline, with user ratings significantly above the agreement threshold. On stability, the paper forms the closed-loop state matrix $A_{\mathrm{CL}}$ in Eq. (19) by combining the discrete-time device model with the lifted Koopman dynamics and the virtual coupling, and shows that all eigenvalues lie inside the unit circle for the case study, establishing asymptotic stability. Because passivity is only a sufficient condition, the authors argue this eigenvalue check covers a broader class of systems and is therefore less conservative. They also report that the Koopman-model VE yields lower RMSE against the rigid-body simulation across repeated trials, which they attribute to its representation of the cubic stiffness term as one component of a linear combination of lifted states rather than as a standalone amplifying nonlinearity.

Load-bearing premise

The load-bearing premise is that the linear Koopman model used to build the closed-loop stability matrix accurately represents the implemented haptic system, which re-lifts its states at every time step and therefore is not purely linear; the paper provides no bound on that approximation error and no stability guarantee covering the re-lifting step.

Editorial extensions

If this is right

  • Virtual-coupling parameters can be selected by imposing the eigenvalue constraint on $A_{\mathrm{CL}}$, turning stability-guaranteed rendering into a linear design problem.
  • The eigenvalue check applies to systems that are not passive, so it is less conservative than passivity-based sufficient conditions.
  • The trained Koopman model is reusable across device models: changing $m_d$, $b_d$, $k_c$, or $b_c$ only reassembles $A_{\mathrm{CL}}$ from the stored $A$ and $B$.
  • For nonlinear environments that require implicit integration, replacing the VE update with the Koopman model reduces per-step cost from $O(k^3)$ to $O(p^2)$.
  • The Koopman-model VE's lower RMSE relative to the rigid-body simulation across repeated trials indicates it is less sensitive to unmodeled structural flexibility of the device than the nonlinear baseline.

Reading between the lines

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

  • The implementation relifts the state at every step, so the rendered system is not literally the linear system whose eigenvalues are checked; a robustness margin that accounts for the residual $r_k$ would make the stability certificate cover the implementation, not just the fitted model.
  • The robustness advantage is demonstrated for a cubic stiffness nonlinearity; for nonsmooth or hysteretic environments the lifting-function approach would need independent validation, and the sensitivity to lifting-function choice already appears in the paper's 8-function versus 9-function comparison.
  • A direct comparison of the largest stable virtual-coupling stiffness allowed by the eigenvalue test versus by passivity-based criteria on the same device would quantify how much conservatism is actually removed.
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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 / 8 minor

Summary. The paper proposes using a learned finite-dimensional Koopman model to represent a nonlinear virtual environment (a Duffing oscillator) in a kinesthetic haptic system. The Koopman model is trained in simulation from the nonlinear VE dynamics, implemented with retraction/relifting at each time step, and compared against the baseline nonlinear-model VE in both a single-user experiment and a multi-user perception study. The authors further use the linear lifted representation to build a closed-loop matrix ACL and claim that an eigenvalue check can certify asymptotic stability of the overall haptic system, that this stability analysis is less conservative than passivity-based methods, and that the Koopman representation is more robust to haptic-device modeling uncertainties than the traditional nonlinear model. The empirical trajectory matches and the multi-user ratings support the representation claim, but the stability and conservativeness claims are not rigorously connected to the actually implemented relifted system.

Significance. If the stability guarantee were rigorously established, the paper would make a useful contribution to haptic rendering by showing that data-driven Koopman models can replace nonlinear VEs and enable linear stability tools in a domain where passivity criteria are typically conservative. The empirical validation is a genuine strength: the Koopman model is tested on experimental trajectories not used in training, the multi-user study provides perceptual evidence, and the appendix reports ten additional trials with consistently lower RMSE for the Koopman-model VE. The main weakness is that the central stability claim is currently only proved for a fitted linear recurrence, not for the relifted implementation that was actually tested; the 'less conservative than passivity' comparison is asserted rather than demonstrated. The robustness claim, while empirically suggestive, is not backed by a formal uncertainty analysis.

major comments (3)
  1. [Section II.B, Eq. (19) and Section III.C] The closed-loop stability analysis in Eq. (19) treats the lifted state as evolving linearly according to ϑ_{k+1} = Aϑ_k + Bu_k. However, Section III.C states that the Koopman-model VE retracts and relifts the states at every step to prevent error propagation in Eq. (7). The actually implemented map is therefore the nonlinear, state-dependent composition ϑ_{k+1} = ψ(f(x_k, u_k)) evaluated through the learned A and B, not the LTI recurrence analyzed in Eq. (19). The residual r_k in Eq. (7) is introduced but never bounded, and no theorem connects the spectral radius of ACL to stability of the relifted closed loop. Consequently, the eigenvalue plot in Fig. 8 certifies only the fitted linear model, not the system that was tested in simulation and experiment. This gap is load-bearing because the abstract's headline stability advantage and the 'less conservative than passivity' claim rest on this eigenvalue check. The authors should either provide a residual bound and a stability theorem that covers the relifted dynamics, or explicitly restrict the stability claim to the linear lifted model and describe the experimental stability evidence as heuristic.
  2. [Section II.B and Section IV.C] The assertion that the proposed eigenvalue check is 'less conservative' than passivity-based methods is not established. The manuscript argues that passivity gives only sufficient conditions while the eigenvalue check applies to a broader class of systems, but this is not a comparison of conservativeness for the same physical haptic system. Passivity-based criteria can certify stability of the actual nonlinear VE under stated assumptions, whereas the eigenvalue check currently certifies only the fitted linear model in Eq. (19), whose relationship to the true relifted nonlinear system is not quantified. To make the claim precise, the authors should provide a formal comparison: for example, a parameter region for the virtual-coupling gains in which passivity is inconclusive but the eigenvalue condition (with a controlled approximation error) still guarantees stability of the implemented system. As written, the comparison to passivity is a logical non-sequitur rather than a proven advantage.
  3. [Section IV.B, Table III and Table V] The claim that the Koopman-model VE is 'more robust to uncertainties in the modeling of the haptic device' is supported only by lower RMSE values relative to a rigid-body simulation baseline across ten experimental trials. This is useful empirical evidence, but RMSE here measures closeness to an idealized rigid-body simulation, not to the measured device behavior, and no uncertainty model or statistical significance test is provided. The explanatory mechanism given in Section IV.B (that the Koopman model does not assign outsized importance to the x_v^3 term) is heuristic. The authors should either frame this as a preliminary empirical observation, add error bars and a formal sensitivity analysis with respect to device-parameter perturbations, or weaken the abstract's robustness claim accordingly.
minor comments (8)
  1. [Abstract] The abstract contains a typo: 'to access stability' should be 'to assess stability'.
  2. [Section III.B] The text reads 'Quanser's hardward-in-the-loop'; 'hardward' should be 'hardware'.
  3. [Section IV.B] The phrase 'sturctural flexibiliy' should be 'structural flexibility', and 'the extend to which' in Section III.D should be 'the extent to which'.
  4. [Section IV.C] The sentence 'the difference in the Lyapunov candidate ... is negative or closed to 0' should read 'close to 0'.
  5. [Section II.C, Eqs. (21)-(22)] The two Lyapunov candidates have inconsistent factors: V_nonlinear uses 1/2 multipliers for the kinetic and coupling terms while V_Koopman omits them. This is not technically wrong, but the inconsistency makes the comparison harder to follow.
  6. [Section IV.A] The simulation match in Fig. 3 is described qualitatively as 'align closely'; a quantitative metric such as the normalized RMSE between the Koopman and nonlinear simulated responses would strengthen the claim.
  7. [Section IV.D] The multi-user study has a small sample (N=11) and the Wilcoxon test is reported only via p < 0.004; reporting the test statistic and the exact p-value would improve reproducibility.
  8. [Section III.C] The phrase 'evolved without external force input' and later 'under free force input' could be misleading; clarify that the VE is released from an initial displacement with zero commanded force input.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation found: the Koopman representation is validated on new experimental data, and the stability analysis is a standard linear-model check rather than an input-to-output equivalence.

full rationale

The paper's central claims do not reduce to their inputs by construction. The Koopman model is identified from simulated data generated by the nonlinear Duffing VE (Section III.C) and then tested on separate experimental trajectories (Section IV.B) and a multi-user perception study (Section IV.D); this is an independent generalization test, not a self-prediction. The closed-loop stability analysis in Section II.B builds the LTI lifted recurrence (15)-(19) and checks the eigenvalues of ACL; this is a standard linear-systems criterion and is not equivalent to the training data. The robustness comparison in Section IV.B uses experimental RMSE values against a simulation baseline, so it is an empirical result rather than a quantity forced by construction. The self-citations (e.g., [22], [25], [27] by Dahdah and Forbes) are methodological and are not used as a load-bearing uniqueness argument, so they do not constitute circularity. One non-circular caveat is worth flagging: the implemented VE explicitly retracts and relifts states at every step to 'prevent error propagation in (7)' (Section III.C), so the implemented update is not the LTI recurrence (15) used to form ACL; absent a bound on the residual r_k in (7), the eigenvalue check in Figure 8 does not by itself certify the relifted implementation. This is a correctness gap, but it is not a circularity because the stability conclusion is not equivalent to the training data or to a fitted parameter by construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on a data-fitted Koopman model, a hand-chosen lifting space, a tuned device damping, and the assumption that the linear lifted model (without relifting) captures the implemented system. These are the main axes of uncertainty.

free parameters (3)
  • Device damping coefficient b_d = Not given numerically in text; tuned to match baseline simulation to experiment (Section III.A)
    Tuned to minimize RMSE between simulated and experimental baseline response; affects simulation and the robustness comparison.
  • Koopman matrices A and B = Estimated from 25 simulated datasets via pykoop (Section III.C)
    The entire stability analysis and simulations depend on these fitted matrices; no confidence bounds are provided.
  • Lifted state dimension and polynomial order (p=9, order 3) = 9 lifted states: x_v, x_v_dot, x_v^2, x_v_dot^2, x_v*x_v_dot, x_v^3, x_v^2*x_v_dot, x_v*x_v_dot^2, x_v_dot^3
    Chosen by hand; the paper notes sensitivity to this choice.
assumptions (5)
  • domain assumption The finite-dimensional Koopman approximation residual r_k in (7) is negligible for simulating the VE and for closed-loop stability analysis.
    The stability analysis uses the learned A and B without considering approximation error; no bound is given for r_k. See Eq. (7) and Section IV.C.
  • domain assumption Inputs to the Koopman model appear linearly, so only states need lifting.
    Stated in Section II.B: 'it is assumed to be sufficient to lift only the states'. This is standard for Koopman with linear inputs but must hold for the Duffing VE.
  • ad hoc to paper The closed-loop system (19) with the learned Koopman model accurately represents the actual haptic system with the relifted implementation.
    The implementation relifts states at each step (Section III.C), while (19) assumes linear evolution; this gap is not addressed.
  • domain assumption The haptic device is well described by the rigid-body mass-damper model (11) for the purposes of stability analysis.
    The paper itself notes structural flexibility and capstan drive are neglected, and the damping is tuned to match experiments.
  • domain assumption Stability of the free-motion closed-loop system (no hand force) is sufficient for the user-interaction stability claim.
    The analysis excludes the human operator dynamics; passivity-based methods typically account for the operator's passivity.

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Cite this review

Pith. "Pith review of Koopman Representation of Nonlinear Virtual Environments in Kinesthetic Haptic Systems." pith.science (2026). https://pith.science/paper/5M7S4PYT

@misc{pith2026260811461,
  author       = {Pith},
  title        = {Pith review of: Koopman Representation of Nonlinear Virtual Environments in Kinesthetic Haptic Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5M7S4PYT}},
  note         = {Machine review of arXiv:2608.11461}
}
read the original abstract

Rendering haptic feedback with nonlinear virtual environments (VEs) is important in many applications that require highly accurate force feedback. This paper considers the use of the Koopman operator to represent a nonlinear VE interacting with a haptic system. Simulation and experimental results demonstrated that the proposed method provides an effective representation of the nonlinear dynamics of a Duffing-oscillator VE. A multi-user study further confirmed this conclusion. In addition, a closed-loop (CL) stability analysis is performed leveraging the Koopman representation of the nonlinear VE to access stability of the overall haptic system. This alternative way of representing nonlinear VEs enables a convenient CL stability analysis that is less conservative than traditional passivity-based methods. Since a linear combination of all lifted states is used to represent the nonlinearity, such representation is also more robust to uncertainties in the modeling of the haptic device than a traditional nonlinear model.

Figures

Figures reproduced from arXiv: 2608.11461 by the authors.

Figure 1
Figure 1. Illustration of haptic system with virtual coupling. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Quanser 2-DOF pantograph [32]. C. Single-User Study A Duffing-oscillator VE with the nonlinear model in (20) with mv = 1 kg, bv = 5 Ns/m, k1 = 100 N/m, k2 = −3 N/m3 was examined in experiments as the baseline. The VE moves in the y-direction as indicated in [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 4
Figure 4. Experimental responses for nonlinear-model and Koopman-model [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: Experimental vs. simulation EE states for nonlinear-model VE. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Experimental vs. simulation EE states for Koopman-model VE. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Histogram of Lyapunov candidate difference for Koopman-model [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Eigenvalues of ACL. the nonlinear-model VE or the Koopman-model VE would require finding alternative Lyapunov candidates, which is challenging. By contrast, the CL analysis with the Koopman￾model VE in the case study shows that all eigenvalues of ACL in (19) lie strict…

Discussion (0). Continue with ORCID to comment.

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