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REVIEW 4 major objections 4 minor 17 references

Learning to Move in Rhythm: Task-Conditioned Motion Policies with Orbital Stability Guarantees

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Orbitally Stable Motion Primitives: learned periodic robot motions with exponential convergence and zero-shot task interpolation.

desk verdict OSMP extends SPDT with real empirical gains, but the headline exponential orbital stability guarantee rests on a gap in the transverse contraction proof and the zero-shot interpolation claim is diluted by the training loss. read the letter →

arxiv 2507.10602 v1 pith:BDNXEREP submitted 2025-07-12 cs.RO cs.AI

Maximilian Stölzle , T. Konstantin Rusch , Zach J. Patterson , Rodrigo Pérez-Dattari , Francesco Stella , Josie Hughes , Cosimo Della Santina , Daniela Rus This is my paper · ORCID
classification cs.ROcs.AI
keywords orbitalstabilitylimitcyclesimitationlearningdynamicmovementprimitivestransversecontractionHopfbifurcationtaskconditioningzero-shotgeneralization
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

Orbitally Stable Motion Primitives (OSMPs) aim to turn a single periodic demonstration into a robot controller with a formal guarantee: every trajectory converges exponentially to the learned limit cycle, not merely asymptotically. The construction maps robot configurations through a bijective, task-conditioned encoder into a latent oscillator built from a supercritical Hopf bifurcation, then pulls the latent velocity back to the robot state space. The paper further claims that one policy, conditioned on a scalar task variable $z$, can reproduce several motions and interpolate smoothly between them, giving consistent zero-shot behavior on unseen tasks inside the training distribution. If these claims hold, rhythmic tasks such as wiping, brooming, and swimming become learnable from very little data while staying robust to disturbances and predictable under human interaction.

What carries the argument

The load-bearing object is the latent-space oscillator in polar coordinates, $$\dot{y}_{\mathrm{pol}} = (\$\alpha$(1-$r^{2}$/$R^{2}$)r,\; f_\omega(\varphi),\;-\$\beta$ y_{3:n}),$$ the polar form of a supercritical Hopf bifurcation: radius $r$ is driven toward $R$ by gain $\alpha$, the transverse coordinates $y_{3:n}$ decay with gain $\beta$, and motion around the cycle is governed by a learned positive angular velocity $f_\omega(\varphi)$. A bijective Euclideanizing-flow encoder $\Psi(x;z)$ -- a composition of invertible coupling layers -- maps the robot configuration $x$ into this latent space, and the policy velocity is the pullback $\dot{x} = f_s(x)J_\Psi^{-1} f_y(\Psi(x;z))$. The convergence proof runs through a transverse contraction metric in polar coordinates, lifted through the polar-to-Cartesian map and then through the diffeomorphism; Theorem 1 packages that chain into almost-global exponential orbital stability. The mechanism for multi-task behavior is conditioning: the encoder depends on a scalar task parameter $z$, and the smooth-conditioning-interpolation loss $\mathcal{L}_{\mathrm{sci}}$ enforces that interpolated latent-space cycles map to linearly interpolated oracle-space cycles.

What would settle it

Evaluate the transverse-contraction inequality of Proposition 1 at a fixed polar state with $y_{3:n} \neq 0$ and $r \neq R$: if the left-hand side is positive there, the exponential contraction claim is not established. In simulation, initialize two nearby trajectories with the same phase but different transverse coordinates; if their transverse separation does not decay at the predicted exponential rate $e^{-\zeta t}$, the central guarantee fails.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a diffeomorphic encoder combined with a supercritical Hopf oscillator in latent space provides both the expressive freedom to reproduce complex periodic trajectories and the structure to prove strong convergence. Theorem 1 states that, whenever the velocity scaling factor is set to $f_s(x)=1$, the OSMP dynamics are transverse contracting almost everywhere, and transverse contraction implies exponential orbital stability: trajectories converge to the periodic orbit at an exponential rate from any initial state outside the center of the cycle. This strengthens the previously asymptotic orbital-stability observation for such systems into an exponential one, under mild architectural assumptions. A second claimed result is that conditioning the encoder on a scalar task variable, with a dedicated interpolation loss, makes the learned limit cycle vary smoothly with the conditioning, so a single policy can transition between behaviors and generalize zero-shot within the training distribution. The paper supports these claims with benchmark rollouts, ablations, and hardware experiments on several robot platforms.

Load-bearing premise

The exponential-convergence guarantee rests on treating a chosen variation between two nearby trajectories as exactly perpendicular to the cycle's motion, but at any state with nonzero transverse coordinates that perpendicularity is only approximate and becomes exact only as the coordinates shrink to zero.

Editorial extensions

If this is right

  • A single demonstration suffices to train a working rhythmic policy, with exponential return to the cycle after disturbances.
  • Because the policy is time-invariant, phase shifts such as a half-period offset do not derail it, unlike classical time-indexed trajectory tracking.
  • Multiple OSMPs can be phase-locked by adjusting only their polar angular velocities, preserving each primitive's spatial motion and convergence properties.
  • A single task-conditioned policy can morph between learned motions and execute unseen motions inside the training distribution without retraining.
  • Exponential transverse contraction makes OSMPs composable: interconnecting them with other contracting systems should preserve stability, supporting modular and hierarchical control.

Reading between the lines

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

  • Editorially, if the exponential guarantee is genuine, OSMPs should remain stable under piecewise-constant switching of the conditioning $z(t)$; the paper names this extension as future work, and it would turn the interpolation result into a discrete mode-switching controller.
  • Beyond the paper, the scalar conditioning channel is a minimal case of a broader idea: replacing $z$ with an observation or language embedding would let the same orbital-stability machinery select behaviors from perception, a direction the paper flags as future work.
  • Since contraction is preserved under interconnection, separate OSMPs synchronized by the phase-feedback law should scale toward high-degree-of-freedom locomotion, with each limb or module kept on its own cycle while only the phase is coupled.
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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

4 major / 4 minor

Summary. The manuscript introduces Orbitally Stable Motion Primitives (OSMPs), a dynamical-system motion policy that combines a task-conditioned bijective encoder (Euclideanizing flow) with latent supercritical Hopf-oscillator dynamics. The paper claims two formal results: asymptotic orbital stability (Theorem 2) and, when the velocity scaling is f_s=1, transverse contraction implying exponential orbital stability (Theorem 1), with proofs relegated to Supplementary Propositions 1 and 2. The method is evaluated against MLP, RNN, LSTM, NODE, Diffusion Policy, iFlow, and SPDT baselines on IROS letters, Drawing2D, image-contour, and turtle-swimming oracles, and is deployed on the UR5, KUKA, Helix soft robot, and Crush turtle robot. Additional contributions include phase synchronization of multiple primitives, online velocity-field shaping, and encoder conditioning with a smooth-interpolation loss (L_sci).

Significance. If the exponential-contraction guarantee were fully established, this would be a meaningful advance over prior orbitally stable DMPs: a single demonstration could yield a policy with a built-in limit cycle and guaranteed exponential convergence, and the empirical study is unusually broad, including real-hardware results, ablations, and an inference-time analysis. The asymptotic orbital stability proof (Theorem 2) is a useful formalization, and the phase-synchronization and online-shaping components are practically valuable. However, the central exponential claim rests on a gap in the transverse-contraction proof, and the zero-shot interpolation claim is overstated because L_sci trains directly on the tested conditioning values. Both issues are localized and potentially fixable, so the appropriate disposition is major revision rather than rejection.

major comments (4)
  1. [Supplementary Proposition 1, Step 2 (Eq. S60)] The orthogonality condition required by Definition 1 is not established. Definition 1 demands that the contraction inequality hold for every variational vector δ with δ^T M f = 0 at each point of the region. The proof instead fixes δ = c(1, 0, 1_{n-2}^T)^T and computes δ^T M_pol f_pol = -c β y_{3:n}, then asserts that the orthogonality condition is fulfilled because -c β y_{3:n} converges uniformly to zero along the flow. That is a statement about the limit as t → ∞, not a pointwise algebraic condition. At any state with y_{3:n} ≠ 0, the chosen δ is not orthogonal to f_pol, so inequality (S61) is applied to a non-admissible perturbation. Consequently, transverse contraction of the polar latent dynamics is not proven, and Theorem 1's exponential orbital stability claim is unsupported as written.
  2. [Supplementary Proposition 2 (Eqs. S66-S67)] Proposition 2 inherits the same orthogonality gap. The Cartesian variational vector δ_y = c(y_1/r, y_2/r, 1_{n-2})^T is obtained from the same non-orthogonal polar vector, and the computation δ_y^T M_y f_y = -c β y_{3:n} again only vanishes along the flow, not at every point of the region. Since Proposition 2 is the bridge from polar to Cartesian coordinates in the proof of Theorem 1, the exponential contraction conclusion is not established in either coordinate system.
  3. [Section 2.1, Theorem 1 and Supplementary Proposition 1 (region mismatch)] The theorem's region X = {x : sqrt(Ψ_1^2 + Ψ_2^2) > 0} corresponds to r > 0, but Proposition 1 states its contraction-rate lower bound only for r ≥ r_ε, with r_ε > 0. For 0 < r < r_ε, no uniform exponential rate is provided, so the claimed 'almost-global exponential orbital stability' does not follow from the supplied proof. The authors should either restrict the exponential claim to a region where a uniform rate is proven or provide a separate argument covering the neighborhood of the origin.
  4. [Section 2.6 and Supplementary Eq. (S12)-(S16)] The 'zero-shot generalization to unseen motion objectives' claim is circular in an important sense. The smooth conditioning interpolation loss L_sci samples conditioning values z̃ uniformly from the convex hull conv(Z) during training and supervises the encoder output at those values against a linear interpolation of the training endpoints. The values marked 'unseen' in Fig. 7D, such as z = -0.5 and z = 0.5, lie inside conv(Z) and are therefore directly optimized during L_sci training. If the authors intend 'within the training distribution' to include all hull points, then the experiment demonstrates interpolation to points that were seen by the loss, not zero-shot generalization; the terminology should be revised or the test values should be genuinely held out from L_sci.
minor comments (4)
  1. [Abstract and Table 1] The abstract's 'consistently outperforming state-of-the-art baselines such as diffusion policies' is stronger than the data support. In the IROS Letters row of Table 1, OSMP has higher trajectory RMSE (0.344) and velocity RMSE (0.895) than both MLP (0.257, 0.630) and Diffusion Policy (0.255, 0.648). The Discussion acknowledges these exceptions, but the abstract and Section 2.2 should be qualified accordingly.
  2. [Supplementary Eq. (S56)] The expression for the smallest eigenvalue condition in the positive-definiteness proof of M_pol is difficult to verify because of the nested parentheses and the notation m_φφ appearing on both sides of the inequality. The derivation should be rewritten more transparently or moved to a supporting calculation.
  3. [Section 4.3, Eq. (6)] The phase-synchronization law multiplies the angular velocity by the factor 1 - k_ps Σ_j sin(δΦ*_ij + iφ - jφ). For sufficiently large k_ps or large phase errors, this factor can become nonpositive, which would violate the assumption ω > 0 used in the latent-dynamics analysis. A bound on k_ps or a projection that preserves positivity should be stated.
  4. [Figure 7D] The labels 'unseen during training' in Fig. 7D should be reconciled with the fact that L_sci samples uniformly over conv(Z); as written, the caption overstates the novelty of the interpolated conditioning values.

Circularity Check

1 steps flagged · score 6.0 of 10

Zero-shot generalization is trained by construction: L_sci samples the test conditioning values and fits the interpolation target, so the reported unseen-task performance is the loss itself; the stability derivation is not circular.

  1. fitted input called prediction [Sec. 2.6; Supplementary 'Smooth Conditioning Interpolation Loss' (Eq. S16); Fig. 7 caption]
    "To ensure the learned policy transitions smoothly across behaviors (e.g., for z∈[−1, 1]), we add a loss term Lsci during training. ... Lsci promotes smooth interpolation between oracles, enabling meaningful zero-shot performance on unseen tasks that fall within the training distribution. Supplement: 'we draw Nsci random conditionings from a uniform distribution: z̃(j)∼U(conv(Z))∈R ... the smooth conditioning interpolation loss can be formulated as Lsci= (1/Nsci) Σ (x̃*(j)−x̃(j))^2.' Fig. 7: 'with z=−0.5 and z=0.5 unseen during training.'"

    The test 'unseen' conditioning values z=−0.5 and z=0.5 lie inside conv(Z)=[zmin,zmax], and L_sci explicitly samples z̃ uniformly from this entire interval during training. The loss target x̃* is the linear interpolation of the inverse encoder at the floor/ceil anchor conditionings, and the loss directly trains the encoder to output exactly that target at every sampled z, including the test values. Therefore the reported zero-shot interpolation is not an emergent prediction at genuinely unseen conditioning points; it is the training objective itself evaluated at those points. The experiment measures fit to the interpolation target, not generalization.

full rationale

Walking the derivation chain: the transverse-contraction claim (Theorem 1, Propositions 1 and 2) is a mathematical proof, not a fitted or self-referential reduction. The orthogonality step at Eq. S60 uses a fixed variational vector whose residual −cβy_{3:n} is asserted to vanish only along the flow; that is a proof correctness risk, not a circularity, and does not enter this score. The change-of-coordinates step cites external contraction theory, and Theorem 2's Lyapunov argument is self-contained. No load-bearing self-citation was found. The one genuine circular step is the zero-shot interpolation claim: L_sci trains the encoder at all z in the convex hull of the training conditionings—including the later-called 'unseen' test values—and its target is precisely the linear-interpolation behavior that is then reported as zero-shot generalization. That portion of the paper's prediction reduces to its training objective by construction, while the stability contributions remain independent.

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

The framework introduces no new physical entities. It does require several hand-set constants and domain assumptions; the most salient are the unproven orthogonality step in the contraction proof and the assumption that linear interpolation in oracle space is a valid task homotopy.

free parameters (6)
  • alpha and beta (latent convergence gains) = alpha = beta = 1
    Positive gains in the Hopf latent dynamics (Eq. 4) that set the contraction rate; chosen by hand, not learned from data.
  • R (latent limit cycle radius) = R = 1 or 0.5
    Radius of the target limit cycle in latent space; fixed by the user rather than estimated.
  • Loss weights zeta_vi, zeta_lcm, zeta_tgd, zeta_er, zeta_vr, zeta_sci = Not fully listed in main text; chosen per dataset
    The training objective (Eq. 5) is a weighted sum of six terms; the ablations in Table S2 show the best combination depends on oracle complexity, so weights and included terms are selected per dataset.
  • Angular velocity network parameters with epsilon_omega = epsilon_omega = 1e-6; MLP hidden dimension 128
    omega(phi) = exp(MLP(ybar)) + epsilon_omega is learned to reproduce complex velocity profiles; the small offset prevents zero angular velocity.
  • Encoder architecture size = 10 to 25 Real NVP blocks depending on task
    The number of coupling layers is tuned by task complexity, affecting expressiveness and distortion of the learned diffeomorphism.
  • Online shaping and synchronization gains = s_f, x_o, s_omega, k_conv, R_sm, sigma_sm, k_ps reported per experiment
    Phase-synchronization gain and velocity-field modulation parameters are set by hand for each deployment.
assumptions (6)
  • standard math The Euclideanizing-flow encoder is a smooth bijection for every trained weight and conditioning z.
    Real NVP coupling layers are analytically invertible, but the paper does not prove that the trained networks remain well-behaved over the whole workspace; this is assumed.
  • standard math The supercritical Hopf latent dynamics have an almost-globally attractive limit cycle for alpha, beta, R > 0.
    Standard dynamical systems result (Strogatz, ref 30); invoked in Section 4.1.2 and used in Theorems 1 and 2.
  • domain assumption Transverse contraction and asymptotic orbital stability are preserved under the learned diffeomorphism Psi.
    The paper cites Manchester and Slotine and others (refs 36-38) for invariance; no proof is given for the specific metric pullback.
  • domain assumption Demonstrations are periodic with a known period P and can be time-aligned to a polar phase anchor.
    Used to define the time-guidance loss (Eq. S8) and to evaluate convergence; real kinesthetic demonstrations often do not close exactly.
  • ad hoc to paper A scalar conditioning z with linear interpolation between training task encodings is a meaningful representation of the task space.
    The L_sci loss (Eq. S16) forces the inverse encoder to follow linear interpolations in oracle space; the paper provides no evidence this matches perceptual or physical task similarity.
  • domain assumption Low-level robot controllers track the OSMP velocity reference closely enough that the orbital stability of the policy is preserved in hardware.
    The paper notes tracking errors on the Helix soft robot and turtle robot come from low-level controllers; hardware results therefore test the combined system, not the OSMP alone.

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Cite this review

Pith. "Pith review of Learning to Move in Rhythm: Task-Conditioned Motion Policies with Orbital Stability Guarantees." pith.science (2026). https://pith.science/paper/BDNXEREP

@misc{pith2026250710602,
  author       = {Pith},
  title        = {Pith review of: Learning to Move in Rhythm: Task-Conditioned Motion Policies with Orbital Stability Guarantees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BDNXEREP}},
  note         = {Machine review of arXiv:2507.10602}
}
read the original abstract

Learning from demonstration provides a sample-efficient approach to acquiring complex behaviors, enabling robots to move robustly, compliantly, and with fluidity. In this context, Dynamic Motion Primitives offer built - in stability and robustness to disturbances but often struggle to capture complex periodic behaviors. Moreover, they are limited in their ability to interpolate between different tasks. These shortcomings substantially narrow their applicability, excluding a wide class of practically meaningful tasks such as locomotion and rhythmic tool use. In this work, we introduce Orbitally Stable Motion Primitives (OSMPs) - a framework that combines a learned diffeomorphic encoder with a supercritical Hopf bifurcation in latent space, enabling the accurate acquisition of periodic motions from demonstrations while ensuring formal guarantees of orbital stability and transverse contraction. Furthermore, by conditioning the bijective encoder on the task, we enable a single learned policy to represent multiple motion objectives, yielding consistent zero-shot generalization to unseen motion objectives within the training distribution. We validate the proposed approach through extensive simulation and real-world experiments across a diverse range of robotic platforms - from collaborative arms and soft manipulators to a bio-inspired rigid-soft turtle robot - demonstrating its versatility and effectiveness in consistently outperforming state-of-the-art baselines such as diffusion policies, among others.

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Reference graph

Works this paper leans on

17 extracted references · 14 canonical work pages

  1. [2]

    (S26) where𝜋=((𝑘1,𝜅 1),..., (𝑘𝑁,𝜅 𝑁)) is commonly referred to as the alignment path of length𝑁 that contains the sequence of index pairs1. In order for𝜋 to be a valid alignment path, it needs to fulfill the following constraints 𝜋1=(𝑘1,𝜅 1)=(1, 1), 𝜋 𝑁 =(𝑘1,𝜅 1)=(𝑁,𝑁), 𝑘𝜄+1−𝑘𝜄∈{ 0, 1}, 𝜅 𝜄+1−𝜅𝜄∈{ 0, 1}, 𝑘 𝜄+1−𝑘𝜄+𝜅𝜄+1−𝜅𝜄≥ 1, (S27) where the first row conta...

    work page

  2. [17]

    inference time

    (S30) The undirected/symmetric version of the Hausdorff distance was used by Zhiet al.(22) for evaluating the similarity between the desired and actual trajectory shape. Iterative Closest Point MED We use an Iterative Closest Point (ICP) algorithm ( 64) to identify the optimal alignment between the two sequences 𝑥d(𝑘)∀ 𝑘∈ 1,...,𝑁 and𝑥(𝜅)∀ 𝜅∈ 1,...,𝑁 conta...

    arXiv 1995

  3. [54]

    optimizer configured with a learning rate of 10 −4, a weight decay of 10 −6, and (𝛽1,𝛽 2) = (0.95, 0.999) and a cosine learning rate scheduler with 10 warmup steps. For this task, we define the observation as the current and last position of the system 𝑜(𝑘) = h 𝑥⊤(𝑘− 1) 𝑥⊤(𝑘) i⊤ ∈ R2𝑛 and the action as the positional state of the system 𝑎 = 𝑥∈ R𝑛. For eac...

    work page

  4. [61]

    Sakoe, S

    H. Sakoe, S. Chiba, Dynamic programming algorithm optimization for spoken word recogni- tion. IEEE Transactions on Acoustics, Speech, and Signal Processing 26 (1), 43–49 (1978), doi:10.1109/TASSP.1978.1163055

    work page arXiv 1978

  5. [62]

    Hausdorff, Grundz¨uge der mengenlehre, vol

    F. Hausdorff, Grundz¨uge der mengenlehre, vol. 7 (von Veit) (1914)

    work page 1914

  6. [63]

    D. P. Huttenlocher, G. A. Klanderman, W. J. Rucklidge, Comparing images using the Hausdorff distance. IEEE Transactions on pattern analysis and machine intelligence 15 (9), 850–863 (1993)

    work page 1993

  7. [64]

    P. J. Besl, N. D. McKay, A Method for Registration of 3-D Shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence14 (2), 239–256 (1992), doi:10.1109/34.121791. 34

    doi:10.1109/34.121791 1992

  8. [65]

    Khatib, A unified approach for motion and force control of robot manipulators: The opera- tional space formulation

    O. Khatib, A unified approach for motion and force control of robot manipulators: The opera- tional space formulation. IEEE Journal on Robotics and Automation 3 (1), 43–53 (1987)

    work page 1987

Show all 17 references
  1. [66]

    Scherzinger, A

    S. Scherzinger, A. Roennau, R. Dillmann, Forward Dynamics Compliance Control (FDCC): A new approach to cartesian compliance for robotic manipulators, in2017 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)(IEEE) (2017), pp. 4568–4575

    2017

  2. [67]

    Urain, D

    J. Urain, D. Tateo, J. Peters, Learning stable vector fields on lie groups. IEEE Robotics and Automation Letters 7 (4), 12569–12576 (2022)

    2022

  3. [68]

    J. Sola, J. Deray, D. Atchuthan, A micro lie theory for state estimation in robotics. arXiv preprint arXiv:1812.01537 (2018)

    2018 arXiv

  4. [69]

    Y. Wang, P. Praveena, D. Rakita, M. Gleicher, Rangedik: An optimization-based robot motion generation method for ranged-goal tasks, in 2023 IEEE International Conference on Robotics and Automation (ICRA) (IEEE) (2023), pp. 9700–9706

    2023

  5. [70]

    Stella, Q

    F. Stella, Q. Guan, C. Della Santina, J. Hughes, Piecewise affine curvature model: a reduced- order model for soft robot-environment interaction beyond pcc, in 2023 IEEE International Conference on Soft Robotics (RoboSoft) (IEEE) (2023), pp. 1–7

    2023

  6. [71]

    Della Santina, A

    C. Della Santina, A. Bicchi, D. Rus, On an improved state parametrization for soft robots with piecewise constant curvature and its use in model based control.IEEE Robotics and Automation Letters 5 (2), 1001–1008 (2020)

    2020

  7. [72]

    I. R. Manchester, Transverse dynamics and regions of stability for nonlinear hybrid limit cycles. IFAC Proceedings Volumes44 (1), 6285–6290 (2011)

    2011

  8. [73]

    W. S. Lohmiller, Contraction analysis of nonlinear systems, Phd thesis, Massachusetts Institute of Technology, Cambridge, MA (1999), available at https://dspace.mit.edu/handle/ 1721.1/9793

    1999

  9. [74]

    K. B. Petersen, M. S. Pedersen, et al., The matrix cookbook. Technical University of Denmark 7 (15), 510 (2008). 35 Acknowledgments Funding: The M.S. was supported under the European Union’s Horizon Europe Program from Project EMERGE - Grant Agreement No. 101070918 and by the ...

    2008

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