{"id":"e028e505-32eb-4963-b4fa-4a4a55492966","arxiv_id":"2608.11461","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors learn a Koopman model of a Duffing-oscillator virtual environment and show it matches the nonlinear model in simulation, hardware, and user perception, while claiming simplified and less conservative stability analysis.","lead":"This paper applies Koopman operator theory, a data-driven method that turns nonlinear dynamics into a linear model in a higher-dimensional space, to represent a nonlinear virtual environment for haptic feedback. The authors validate the representation on a Duffing oscillator in simulation and with a haptic device, and argue it makes stability analysis easier and less conservative than passivity-based methods.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stability check via ACL in Eq. (19) uses a purely linear recurrence, while the implemented Koopman VE relifts states every step; without a residual bound, the eigenvalue test may not certify stability of the actual system.","rationale":"The central claim has two parts: an effective representation (supported by simulation, single-user experiments, and a multi-user study) and a stability analysis enabled by the Koopman linearization (claimed to be less conservative than passivity-based methods). The second part is the more novel theoretical contribution. The reader's weakest assumption correctly identifies that the ACL analysis applies to the linear recurrence, not the relifted implementation. I reviewed the full text and found no bound on r_k in Eq. (7) and no treatment of the relifting operation in the stability analysis; Section III.C states relifting was used, and Section IV.C uses the eigenvalue check as evidence of asymptotic stability of the closed-loop system. This is a genuine gap: the analyzed system is formally different from the tested system. The robustness claim (lower RMSE for the Koopman VE) is empirical and could also benefit from statistical support, but it is not the most load-bearing issue because the paper's stated novelty is the stability analysis. I therefore agree with the reader's assessment and would keep the verdict CONDITIONAL, asking the authors to either prove stability for the relifted system (e.g., via a residual bound and small-gain condition) or explicitly restrict the stability claim to the non-relifted Koopman model. The representation results deserve credit: they are reproducible, use the public pykoop package, and the user study provides subjective validation. The concern does not invalidate the representation contribution, but it does mean the headline stability guarantee is unverified for the actual implementation.","tokens_in":11540,"tokens_out":5465,"duration_ms":49029,"concrete_test":"Re-implement both the linear recurrence (15) and the relifted Koopman VE used in Section III.C in simulation. For a fixed device model and a grid of virtual coupling parameters (k_c, b_c) spanning the unit-circle boundary of ACL, simulate both systems from perturbed initial conditions and record whether trajectories remain bounded. If any parameter with all ACL eigenvalues inside the unit circle yields unbounded growth in the relifted system (or vice versa), the eigenvalue criterion is invalid for the implementation. As a secondary check, compute the empirical residual norm ||r_k|| on the training/validation data and compare it to the norms of A and B to see whether a small-gain robustness argument could close the gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central stability claim is that closed-loop stability can be assessed by checking eigenvalues of ACL in Eq. (19). This is valid for the linear recurrence (15)-(19) in which the lifted state evolves linearly. However, the experimental and simulation implementation in Section III.C explicitly retracts and relifts the states at each step to prevent error propagation in (7). The actual VE update is therefore not the LTI recurrence ϑ_{k+1} = Aϑ_k + Bu_k, but a nonlinear state-dependent update involving the projection ϑ(x_k). ACL does not describe this relifted system. 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 implementation. Consequently, the eigenvalue result in Fig. 8 does not establish asymptotic stability of the implemented haptic system. Since the claimed advantage over passivity-based methods rests on this eigenvalue check, the gap is load-bearing. The representation accuracy results are supported by experiments and a user study, but the headline stability guarantee is not delivered for the system that was actually tested.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1731,"tokens_out":2355,"duration_ms":66134,"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":[{"comment":"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.","section":"Section II.B, Eq. (19) and Section III.C"},{"comment":"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.","section":"Section II.B and Section IV.C"},{"comment":"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.","section":"Section IV.B, Table III and Table V"}],"minor_comments":[{"comment":"The abstract contains a typo: 'to access stability' should be 'to assess stability'.","section":"Abstract"},{"comment":"The text reads 'Quanser's hardward-in-the-loop'; 'hardward' should be 'hardware'.","section":"Section III.B"},{"comment":"The phrase 'sturctural flexibiliy' should be 'structural flexibility', and 'the extend to which' in Section III.D should be 'the extent to which'.","section":"Section IV.B"},{"comment":"The sentence 'the difference in the Lyapunov candidate ... is negative or closed to 0' should read 'close to 0'.","section":"Section IV.C"},{"comment":"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.","section":"Section II.C, Eqs. (21)-(22)"},{"comment":"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.","section":"Section IV.A"},{"comment":"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.","section":"Section IV.D"},{"comment":"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.","section":"Section III.C"}],"recommendation":"major_revision","confidential_remarks":"The experimental and user-study portions are solid and the representation results are credible. The main concern is the stability analysis, which does not cover the relifted implementation actually tested. I would encourage the editor to require the authors to either add a rigorous residual-bound or robust-stability argument connecting ACL to the implemented system, or to explicitly soften the stability and passivity-comparison claims. The 'less conservative than passivity' claim needs a concrete parameter-region comparison before it can be accepted. The paper would also benefit from citing recent results on finite-dimensional Koopman approximation error and on data-driven stability certificates. A revision that addresses the stability gap and reframes the robustness claim would be suitable for this venue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the representation work is genuinely solid and the experiments back it up, but the headline stability guarantee is computed on a different system than the one actually implemented. That gap needs to be acknowledged and fixed before you trust the \"less conservative than passivity\" claim.\n\nWhat's new: applying Koopman/EDMD to nonlinear virtual environments for haptics is new in this context, and the paper does a clean job of training a polynomial-lifted model for a Duffing-oscillator VE and showing it tracks the nonlinear model in simulation and on a Quanser pantograph. The multi-user study (median 9, p<0.004) supports that the Koopman VE feels like the baseline. That part is credible. The stability analysis via eigenvalues of ACL is natural and simple, which is a real practical advantage over Lyapunov or passivity approaches if it works.\n\nThe soft spot: the stability analysis uses the linear recurrence in Eq. (19), where the lifted state evolves linearly. But the actual implementation in Section III.C retracts and relifts states at every step, so the effective VE update is a composition of retract-lift with the Koopman map, not the LTI recurrence. ACL is the wrong model for that. The residual r_k is introduced but never bounded, and no theorem connects the spectral radius of ACL to stability of the relifted system. So Fig. 8 is an eigenvalue plot for a different system, and the claim that this is \"less conservative than passivity\" overreaches—passivity may be conservative, but your eigenvalue test does not necessarily certify the real system either. That is the load-bearing issue.\n\nSmaller concern: the robustness-to-device-uncertainty claim is supported by consistently lower RMSE across ten trials, but there is no error analysis or statistical test on those RMSE differences; the numbers are close. The argument about the cubic term being diluted in the lifted space is plausible but informal.\n\nBottom line: this is a useful applied paper and deserves peer review. I would recommend major revision: either reconcile the stability analysis with the relifted implementation (e.g., via a residual bound and a robust stability argument) or clearly label the eigenvalue check as a design heuristic and soften the \"less conservative\" wording. The representation results can stand on their own.","headline":"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.","tokens_in":12286,"tokens_out":3454,"would_cite":false,"duration_ms":28020,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Koopman operator","haptic rendering","nonlinear virtual environment","closed-loop stability","Duffing oscillator","passivity","data-driven model","eigenvalue analysis"],"falsifier":"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.","tokens_in":11374,"feed_emoji":"🕹️","tokens_out":9321,"duration_ms":117414,"temperature":0.7,"pith_summary":"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.","feed_headline":"Learned linear model turns nonlinear haptics into an eigenvalue check","feed_subtitle":"A data-driven linear model of the virtual environment replaces passivity conditions with a direct eigenvalue test.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Koopman-with-input formulation used to write the lifted VE model as $\\vartheta_{k+1}=A\\vartheta_k+Bu_k$.","marker":"[21]"},{"why":"Provides the closed-loop Koopman approximation approach and the convention that original states appear first in the lifted vector.","marker":"[22]"},{"why":"Defines the EDMD/DMD least-squares fit of the Koopman operator and the residual term in the lifted dynamics.","marker":"[24]"},{"why":"The software package used to train the Koopman model from simulated Duffing-oscillator data.","marker":"[27]"},{"why":"The passivity-based haptic stability framework that the proposed eigenvalue test aims to beat as less conservative.","marker":"[9]"},{"why":"A prior guaranteed-stability result for nonlinear VEs that relies on passivity and thus motivates a less conservative alternative.","marker":"[12]"},{"why":"Background for Lyapunov and passivity theory supporting the claim that passivity gives sufficient, not necessary, stability conditions.","marker":"[28]"},{"why":"Provides the device effective mass used in simulations and in assembling the closed-loop matrix.","marker":"[30]"},{"why":"Basis for the retraction-and-relifting implementation that prevents error propagation in the Koopman update.","marker":"[33]"},{"why":"The Wilcoxon signed-rank test used to confirm that user similarity ratings are significantly above the agreement threshold.","marker":"[34]"}],"fun_headline_variants":["Koopman lift turns nonlinear haptics into a linear stability check","Eigenvalue test replaces passivity for nonlinear haptic stability","Data-driven linear model for nonlinear haptic feedback","Nonlinear haptics get a Koopman linearization for stability","Koopman model turns virtual environment nonlinearity into a linear system"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Koopman lift turns nonlinear haptics into a linear stability check","Eigenvalue test replaces passivity for nonlinear haptic stability","Data-driven linear model for nonlinear haptic feedback","Nonlinear haptics get a Koopman linearization for stability","Koopman model turns virtual environment nonlinearity into a linear system"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000446,"raw_usage":{"total_tokens":2254,"prompt_tokens":945,"completion_tokens":1309,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":561,"completion_tokens_details":{"reasoning_tokens":1221}},"tokens_in":561,"tokens_out":1309,"duration_ms":10406,"temperature":1.0,"reasoning_tokens":1221,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:12:09.406872+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Generalizing koopman theory to allow for inputs and control,","cited_arxiv_id":null,"evidence_quote":"Supplies the Koopman-with-input formulation used to write the lifted VE model as $\\vartheta_{k+1}=A\\vartheta_k+Bu_k$."},{"cited_title":"Closed-loop Koopman operator ap- proximation,","cited_arxiv_id":null,"evidence_quote":"Provides the closed-loop Koopman approximation approach and the convention that original states appear first in the lifted vector."},{"cited_title":"Dahdah and J","cited_arxiv_id":null,"evidence_quote":"The software package used to train the Koopman model from simulated Duffing-oscillator data."},{"cited_title":"Stable haptic interaction with virtual environments,","cited_arxiv_id":null,"evidence_quote":"The passivity-based haptic stability framework that the proposed eigenvalue test aims to beat as less conservative."},{"cited_title":"Guaranteed stability of haptic systems with nonlinear virtual environments,","cited_arxiv_id":null,"evidence_quote":"A prior guaranteed-stability result for nonlinear VEs that relies on passivity and thus motivates a less conservative alternative."},{"cited_title":"Dynamics of coupled haptic sys- tems,","cited_arxiv_id":null,"evidence_quote":"Provides the device effective mass used in simulations and in assembling the closed-loop matrix."},{"cited_title":"Learning data- driven stable koopman operators,","cited_arxiv_id":null,"evidence_quote":"Basis for the retraction-and-relifting implementation that prevents error propagation in the Koopman update."},{"cited_title":"Wilcoxon signed-rank test,","cited_arxiv_id":null,"evidence_quote":"The Wilcoxon signed-rank test used to confirm that user similarity ratings are significantly above the agreement threshold."}],"review_version":1}