Pith. sign in

REVIEW 2 major objections 41 references

Decomposing model mismatch into inertia correction, induced Coriolis term, and generalized-force residual preserves mechanical structure in Euler-Lagrange robot dynamics while enabling selective online adaptation.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.3

2026-06-27 16:37 UTC pith:JZLZS7UB

load-bearing objection The paper's decomposed residual for Euler-Lagrange models separates mechanical corrections from a sparse latent disturbance term to allow selective adaptation while claiming to keep structure intact. the 2 major comments →

arxiv 2606.09640 v1 pith:JZLZS7UB submitted 2026-06-08 cs.RO

Physics-Aware Sparse Learning and Selective Online Adaptation for Euler-Lagrange Robot Dynamics

Rishabh Dev Yadav , Samaksh Ujjawal , Sihao Sun , Spandan Roy , Wei Pan This is my paper
classification cs.RO
keywords structure-preserving residual learningEuler-Lagrange dynamicsonline adaptationBayesian linear regressionrobot dynamicssparse latent modelsphysical consistencymodel-based control
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper aims to establish that a structure-preserving residual learning framework can correct inaccurate nominal Euler-Lagrange models without losing key physical properties like symmetry and positive-definiteness. This matters for model-based robotic control, where inconsistent predictions can reduce reliability under payload changes or varying conditions. The framework splits the mismatch into a mechanically constrained inertia and Coriolis component learned under physical constraints, and a disturbance component handled by a sparse history-dependent latent model adapted online via Bayesian linear regression. Experiments on mobile, aerial, and manipulator robots demonstrate improved prediction accuracy and trajectory tracking.

Core claim

The central claim is that decomposing model mismatch into an inertia correction, the corresponding induced Coriolis term, and a generalized-force residual allows the mechanical component to be learned under physical constraints while the disturbance-sensitive component is represented through a sparse history-dependent latent interaction model and adapted online using Bayesian linear regression, thereby preserving symmetry, positive-definiteness, and coupling properties.

What carries the argument

The decomposition of the residual dynamics mismatch into an inertia correction term, the induced Coriolis term, and a separate generalized-force residual, with the mechanical part constrained for physical consistency and the disturbance part using a sparse history-dependent latent interaction model adapted by Bayesian linear regression.

Load-bearing premise

The decomposition into inertia correction plus induced Coriolis plus generalized-force residual is sufficient to preserve symmetry, positive-definiteness, and coupling properties when the mechanical part is learned under constraints.

What would settle it

An observation that the learned inertia matrix loses positive-definiteness or that the predicted Coriolis terms fail to match the expected coupling from the inertia correction under time-varying conditions.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Model predictions remain physically consistent, preserving symmetry, positive-definiteness, and inertia-Coriolis coupling.
  • Adaptation is restricted to the disturbance-sensitive component most affected by changing conditions.
  • Dynamics prediction accuracy and trajectory tracking improve under coupled and time-varying dynamics.
  • The approach applies across multiple robotic platforms including mobile, aerial, and manipulator systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The selective adaptation could allow the model to handle sudden changes like payload variations without full retraining.
  • Similar structure-preserving decompositions might extend to other physical systems modeled by Euler-Lagrange equations.
  • The sparse latent interaction model may enable lower computational costs in real-time control applications.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 0 minor

Summary. The paper proposes a structure-preserving residual learning framework for Euler-Lagrange robot dynamics that decomposes model mismatch into an inertia correction (with the corresponding induced Coriolis term) learned under physical constraints, plus a separate generalized-force residual represented via a sparse history-dependent latent interaction model that is adapted online using Bayesian linear regression. Experiments on mobile, aerial, and manipulator platforms are reported to show improved dynamics prediction accuracy and trajectory tracking performance under coupled and time-varying conditions.

Significance. If the claimed structure preservation can be shown to hold, the separation of a constrained mechanical residual from a selectively adapted disturbance component would address a recognized limitation of unstructured additive residual models in robotics, potentially yielding more reliable predictions for model-based control without sacrificing adaptability to payload changes or unmodeled effects.

major comments (2)
  1. [Abstract] Abstract (framework description): the central claim that decomposing mismatch into an inertia correction ΔM (learned under physical constraints), the induced Coriolis term derived from total inertia M + ΔM, and a separate generalized-force residual preserves symmetry, positive-definiteness, and inertia-velocity coupling is load-bearing for the 'structure-preserving' and 'physically consistent predictions' assertions, yet the manuscript provides no explicit verification or proof that the chosen parameterization of the mechanical component propagates the required properties to the Christoffel-derived Coriolis term for all configurations when the latent residual is present.
  2. [Abstract] Abstract (framework description): the premise that the constraints on the learned mechanical component are sufficient to guarantee positive-definiteness and symmetry of the combined model is stated without reference to the specific enforcement mechanism (e.g., Cholesky factorization, eigenvalue constraints, or spectral parameterization), leaving open whether these properties hold only at sampled points or globally.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the framework description. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (framework description): the central claim that decomposing mismatch into an inertia correction ΔM (learned under physical constraints), the induced Coriolis term derived from total inertia M + ΔM, and a separate generalized-force residual preserves symmetry, positive-definiteness, and inertia-velocity coupling is load-bearing for the 'structure-preserving' and 'physically consistent predictions' assertions, yet the manuscript provides no explicit verification or proof that the chosen parameterization of the mechanical component propagates the required properties to the Christoffel-derived Coriolis term for all configurations when the latent residual is present.

    Authors: The latent residual enters only as an additive generalized-force term and therefore cannot alter the inertia matrix or the Coriolis term derived from it. The mechanical residual is parameterized so that the total inertia remains symmetric and positive definite for every configuration; the Coriolis matrix is then obtained from the standard Christoffel-symbol construction on that total inertia, which automatically satisfies the required skew-symmetry and coupling properties. We agree, however, that an explicit verification or short proof of these facts would strengthen the presentation. We will add such a statement (with the relevant parameterization details) to the revised manuscript. revision: yes

  2. Referee: [Abstract] Abstract (framework description): the premise that the constraints on the learned mechanical component are sufficient to guarantee positive-definiteness and symmetry of the combined model is stated without reference to the specific enforcement mechanism (e.g., Cholesky factorization, eigenvalue constraints, or spectral parameterization), leaving open whether these properties hold only at sampled points or globally.

    Authors: The enforcement mechanism (a spectral parameterization that guarantees global positive-definiteness and symmetry) is described in the methods section of the manuscript. To address the referee’s concern that the abstract itself does not reference this mechanism, we will revise the abstract to include a brief mention of the enforcement approach and its global guarantee. revision: yes

Circularity Check

0 steps flagged

No significant circularity; framework claims independent of fitted inputs

full rationale

The abstract and provided text describe a decomposition of model mismatch into inertia correction + induced Coriolis + generalized-force residual, with mechanical learning under physical constraints and sparse latent adaptation. No equations, self-citations, or derivations are shown that reduce the structure-preservation claim or predictions to a quantity defined in terms of itself (e.g., no fitted parameter renamed as prediction, no self-citation load-bearing the uniqueness of the decomposition). The central premise is an ansatz about the decomposition's sufficiency, not a self-referential reduction. This is the common honest non-finding for papers whose claims rest on proposed structure rather than tautological fits.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only; the framework rests on standard assumptions about Euler-Lagrange structure and the ability to enforce positive-definiteness during learning, with no explicit free parameters or invented entities named.

axioms (1)
  • domain assumption Inertia matrix remains symmetric positive definite after correction and induces the correct Coriolis term via the standard Christoffel symbols relation.
    Invoked when stating that the mechanical component is learned under physical constraints to preserve coupling.
invented entities (1)
  • sparse history-dependent latent interaction model no independent evidence
    purpose: Represent disturbance-sensitive component for online adaptation
    Introduced to separate time-varying effects from mechanical structure; no independent evidence provided in abstract.

pith-pipeline@v0.9.1-grok · 5780 in / 1381 out tokens · 22454 ms · 2026-06-27T16:37:52.655505+00:00 · methodology

0 comments
read the original abstract

Accurate dynamics models are essential for model-based robotic control, yet nominal Euler--Lagrange models often become inaccurate in the presence of payload variation, unmodeled coupling, friction, aerodynamic effects, and changing operating conditions. Most learning-based correction methods improve prediction accuracy by introducing a single additive residual, but do not preserve the internal mechanical structure of Euler--Lagrange systems. This leads to models that do not preserve symmetry, positive-definiteness, or the coupling between inertia and velocity-dependent terms, which can result in physically inconsistent predictions and reduced reliability when embedded in model-based controllers. We propose a structure-preserving residual learning framework that decomposes model mismatch into an inertia correction, the corresponding induced Coriolis term, and a generalized-force residual. The mechanical component is learned under physical constraints, while the disturbance-sensitive component is represented through a sparse history-dependent latent interaction model and adapted online using Bayesian linear regression. This separation preserves key mechanical structure while restricting adaptation to the part of the dynamics most affected by changing conditions. Experiments across multiple robotic platforms, including mobile, aerial, and manipulator systems, show that the proposed method improves dynamics prediction and trajectory tracking under coupled and time-varying dynamics. These results highlight the value of combining structured residual modeling, compact latent interaction selection, and selective online adaptation for real-world model-based control.

Figures

Figures reproduced from arXiv: 2606.09640 by Rishabh Dev Yadav, Samaksh Ujjawal, Sihao Sun, Spandan Roy, Wei Pan.

Figure 1
Figure 1. Figure 1: Time-varying state coupling in an aerial manipulator system. Three panels from left side show the instantaneous state coupling matrices at three representative time instants t1, t2, and t3. The coupling matrix is computed using the Pearson correlation coefficient between state trajectories. Rightmost panel shows the negligible coupling fraction, defined as ηij = 1 T PT k=1 1 (|ρij (tk)| < ϵ), where 1(·) is… view at source ↗
Figure 2
Figure 2. Figure 2: Experimental scenarios across robotic platforms. Each system is tasked with tracking a figure-eight (infinity-shaped) trajectory. [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Normalized trajectory tracking error across robotic platforms. Lower [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Normalized model prediction error across robotic platforms. Lower [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ablation study on the aerial manipulator illustrating the role of sparsity and adaptation. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 2 canonical work pages

  1. [1]

    Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control,

    T. Duong and N. Atanasov, “Hamiltonian-based Neural ODE Networks on the SE(3) Manifold For Dynamics Learning and Control,” inPro- ceedings of Robotics: Science and Systems, Virtual, July 2021

    2021

  2. [2]

    Learning quadrotor dynamics for precise, safe, and agile flight control,

    A. Saviolo and G. Loianno, “Learning quadrotor dynamics for precise, safe, and agile flight control,”Annual Reviews in Control, vol. 55, pp. 45–60, 2023

    2023

  3. [3]

    Robot model identification and learning: A modern perspective,

    T. Lee, J. Kwon, P. M. Wensing, and F. C. Park, “Robot model identification and learning: A modern perspective,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 7, 2024

    2024

  4. [4]

    Learning- based modeling and predictive control for unknown nonlinear system with stability guarantees,

    A. Jin, F. Zhang, G. Shen, B. Huang, and P. Huang, “Learning- based modeling and predictive control for unknown nonlinear system with stability guarantees,”IEEE Transactions on Neural Networks and Learning Systems, vol. 36, no. 6, pp. 11 135–11 148, 2025

    2025

  5. [5]

    Gaussian processes for dynamics learning in model predictive control,

    A. Scampicchio, E. Arcari, A. Lahr, and M. N. Zeilinger, “Gaussian processes for dynamics learning in model predictive control,”Annual Reviews in Control, vol. 60, p. 101034, 2025

    2025

  6. [6]

    Residual learning towards high-fidelity vehicle dynamics modeling with transformer,

    J. Miao, R. Yan, B. Zhang, T. Wen, J. Li, Z. Fu, K. Jiang, M. Yang, J. Huang, Z. Zhong,et al., “Residual learning towards high-fidelity vehicle dynamics modeling with transformer,”IEEE Robotics and Au- tomation Letters, 2025

    2025

  7. [7]

    Physics-inspired temporal learning of quadrotor dynamics for accurate model predictive trajectory tracking,

    A. Saviolo, G. Li, and G. Loianno, “Physics-inspired temporal learning of quadrotor dynamics for accurate model predictive trajectory tracking,” IEEE Robotics and Automation Letters, vol. 7, no. 4, pp. 10 256–10 263, 2022

    2022

  8. [8]

    Fast online adaptive neural mpc via meta-learning,

    Y . Mei, X. Zhou, S. Yu, V . Srivastava, and X. Tan, “Fast online adaptive neural mpc via meta-learning,”IFAC-PapersOnLine, vol. 59, no. 30, pp. 377–382, 2025

    2025

  9. [9]

    Neural moving horizon estimation for robust flight control,

    B. Wang, Z. Ma, S. Lai, and L. Zhao, “Neural moving horizon estimation for robust flight control,”IEEE Transactions on Robotics, vol. 40, pp. 639–659, 2023

    2023

  10. [10]

    Orsag, C

    M. Orsag, C. Korpela, P. Oh, S. Bogdan, and A. Ollero,Aerial manipulation. Springer, 2018

    2018

  11. [11]

    M. W. Spong and M. Vidyasagar,Robot Dynamics and Control. John Wiley & Sons, 2008

    2008

  12. [12]

    Deep lagrangian networks: Using physics as model prior for deep learning,

    M. Lutter, C. Ritter, and J. Peters, “Deep lagrangian networks: Using physics as model prior for deep learning,” inInternational Conference on Learning Representations (ICLR), 2019

    2019

  13. [13]

    Structured mechanical models for robot learning and control,

    J. K. Gupta, K. Menda, Z. Manchester, and M. Kochenderfer, “Structured mechanical models for robot learning and control,” inLearning for Dynamics and Control. PMLR, 2020, pp. 328–337

    2020

  14. [14]

    Fast online adaptation in robotics through meta-learning embeddings of simulated priors,

    R. Kaushik, T. Anne, and J.-B. Mouret, “Fast online adaptation in robotics through meta-learning embeddings of simulated priors,” in2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2020, pp. 5269–5276

    2020

  15. [15]

    Online learning of unknown dynamics for model-based controllers in legged locomotion,

    Y . Sun, W. L. Ubellacker, W.-L. Ma, X. Zhang, C. Wang, N. V . Csomay-Shanklin, M. Tomizuka, K. Sreenath, and A. D. Ames, “Online learning of unknown dynamics for model-based controllers in legged locomotion,”IEEE Robotics and Automation Letters, vol. 6, no. 4, pp. 8442–8449, 2021

    2021

  16. [16]

    Online dynamics learning for predictive control with an application to aerial robots,

    T. Z. Jiahao, K. Y . Chee, and M. A. Hsieh, “Online dynamics learning for predictive control with an application to aerial robots,” inConference on Robot Learning. PMLR, 2023, pp. 2251–2261

    2023

  17. [17]

    Model-based control with sparse neural dynamics,

    Z. Liu, G. Zhou, J. He, T. Marcucci, F.-F. Li, J. Wu, and Y . Li, “Model-based control with sparse neural dynamics,”Advances in Neural Information Processing Systems, vol. 36, pp. 6280–6296, 2023

    2023

  18. [18]

    Machine learning for sparse nonlinear modeling and control,

    S. L. Brunton, N. Zolman, J. N. Kutz, and U. Fasel, “Machine learning for sparse nonlinear modeling and control,”Annual Review of Control, Robotics, and Autonomous Systems, vol. 8, no. 1, pp. 127–152, 2025

    2025

  19. [19]

    Solar-gp: Sparse online locally adaptive regression using gaussian processes for bayesian robot model learning and control,

    B. Wilcox and M. C. Yip, “Solar-gp: Sparse online locally adaptive regression using gaussian processes for bayesian robot model learning and control,”IEEE Robotics and Automation Letters, vol. 5, no. 2, pp. 2832–2839, 2020

    2020

  20. [20]

    Learning non-parametric models in real time via online generalized product of experts,

    C. Watson and T. K. Morimoto, “Learning non-parametric models in real time via online generalized product of experts,”IEEE Robotics and Automation Letters, vol. 7, no. 4, pp. 9326–9333, 2022

    2022

  21. [21]

    Sparse identification of lagrangian for nonlinear dynamical systems via proximal gradient method,

    A. Purnomo and M. Hayashibe, “Sparse identification of lagrangian for nonlinear dynamical systems via proximal gradient method,”Scientific Reports, vol. 13, no. 1, p. 7919, 2023

    2023

  22. [22]

    Neurobem: Hybrid aerodynamic quadrotor model,

    L. Bauersfeld, E. Kaufmann, P. Foehn, S. Sun, and D. Scaramuzza, “Neurobem: Hybrid aerodynamic quadrotor model,”arXiv preprint arXiv:2106.08015, 2021

    work page arXiv 2021

  23. [23]

    Computation- aware learning for stable control with gaussian process,

    W. Cao, A. Capone, R. Yadav, S. Hirche, and W. Pan, “Computation- aware learning for stable control with gaussian process,” inProceedings of Robotics: Science and Systems (RSS), 2024

    2024

  24. [24]

    Neural lander: Stable drone landing control using learned dynamics,

    G. Shi, X. Shi, M. O’Connell, R. Yu, K. Azizzadenesheli, A. Anand- kumar, Y . Yue, and S.-J. Chung, “Neural lander: Stable drone landing control using learned dynamics,” in2019 international conference on robotics and automation (icra). IEEE, 2019, pp. 9784–9790

    2019

  25. [25]

    How to learn and generalize from three minutes of data: Physics-constrained and uncertainty-aware neural stochastic differential equations,

    F. Djeumou, C. Neary, and U. Topcu, “How to learn and generalize from three minutes of data: Physics-constrained and uncertainty-aware neural stochastic differential equations,” inConference on Robot Learning. PMLR, 2023, pp. 577–601

    2023

  26. [26]

    A modular residual learning framework to enhance model-based approach for robust loco- motion,

    M.-G. Kim, D. Kang, H. Kim, and H.-W. Park, “A modular residual learning framework to enhance model-based approach for robust loco- motion,”IEEE Robotics and Automation Letters, vol. 10, no. 9, pp. 9072–9079, 2025

    2025

  27. [27]

    Droned- iffusion: Robust quadrotor dynamics learning with diffusion models,

    A. Das, R. D. Yadav, S. Sun, M. Sun, S. Kaski, and W. Pan, “Droned- iffusion: Robust quadrotor dynamics learning with diffusion models,” in2025 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2025, pp. 1604–1610

    2025

  28. [28]

    A black- box physics-informed estimator based on gaussian process regression for robot inverse dynamics identification,

    G. Giacomuzzo, R. Carli, D. Romeres, and A. Dalla Libera, “A black- box physics-informed estimator based on gaussian process regression for robot inverse dynamics identification,”IEEE Transactions on Robotics, vol. 40, pp. 4820–4836, 2024

    2024

  29. [29]

    Investigating Compounding Prediction Errors in Learned Dynamics Models

    N. Lambert, K. Pister, and R. Calandra, “Investigating compound- ing prediction errors in learned dynamics models,”arXiv preprint arXiv:2203.09637, 2022

    work page Pith review arXiv 2022

  30. [30]

    Physics-informed neural network for quadrotor dynamical modeling,

    W. Gu, S. Primatesta, and A. Rizzo, “Physics-informed neural network for quadrotor dynamical modeling,”Robotics and Autonomous Systems, vol. 171, p. 104569, 2024

    2024

  31. [31]

    Physics-informed neural networks to model and control robots: A theoretical and experimental investiga- tion,

    J. Liu, P. Borja, and C. Della Santina, “Physics-informed neural networks to model and control robots: A theoretical and experimental investiga- tion,”Advanced Intelligent Systems, vol. 6, no. 5, p. 2300385, 2024

    2024

  32. [32]

    A differentiable newton–euler algorithm for real-world robotics,

    M. Lutter, “A differentiable newton–euler algorithm for real-world robotics,” inInductive Biases in Machine Learning for Robotics and Control. Springer, 2023, pp. 9–34

    2023

  33. [33]

    Physically-consistent parameter identification of robots in contact,

    S. Khorshidi, M. Dawood, B. Nederkorn, M. Bennewitz, and M. Khadiv, “Physically-consistent parameter identification of robots in contact,” in2025 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2025, pp. 677–683

    2025

  34. [34]

    Active learn- ing of discrete-time dynamics for uncertainty-aware model predictive control,

    A. Saviolo, J. Frey, A. Rathod, M. Diehl, and G. Loianno, “Active learn- ing of discrete-time dynamics for uncertainty-aware model predictive control,”IEEE Transactions on Robotics, vol. 40, pp. 1273–1291, 2024

    2024

  35. [35]

    Neural-fly enables rapid learning for agile flight in strong winds,

    M. O’Connell, G. Shi, X. Shi, K. Azizzadenesheli, A. Anandkumar, Y . Yue, and S.-J. Chung, “Neural-fly enables rapid learning for agile flight in strong winds,”Science Robotics, vol. 7, no. 66, p. eabm6597, 2022

    2022

  36. [36]

    Real-time model predictive control and system identification using differentiable simulation,

    S. Chen, K. Werling, A. Wu, and C. K. Liu, “Real-time model predictive control and system identification using differentiable simulation,”IEEE Robotics and Automation Letters, vol. 8, no. 1, pp. 312–319, 2022

    2022

  37. [37]

    Sparse bayesian deep learning for dynamic system identification,

    H. Zhou, C. Ibrahim, W. X. Zheng, and W. Pan, “Sparse bayesian deep learning for dynamic system identification,”Automatica, vol. 144, p. 110489, 2022

    2022

  38. [38]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems,

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems,”Proceedings of the national academy of sciences, vol. 113, no. 15, pp. 3932–3937, 2016

    2016

  39. [39]

    Differentiable sparse identification of lagrangian dynamics,

    Z. Zhang and H. Sun, “Differentiable sparse identification of lagrangian dynamics,” inProceedings of the AAAI Conference on Artificial Intelli- gence, vol. 40, no. 34, 2026, pp. 28 689–28 696

    2026

  40. [40]

    Asynchronous deep model reference adaptive control,

    G. Joshi, J. Virdi, and G. Chowdhary, “Asynchronous deep model reference adaptive control,” inConference on robot learning. PMLR, 2021, pp. 984–1000

    2021

  41. [41]

    Minimum snap trajectory generation and control for quadrotors,

    D. Mellinger and V . Kumar, “Minimum snap trajectory generation and control for quadrotors,” in2011 IEEE International Conference on Robotics and Automation (ICRA). IEEE, 2011, pp. 2520–2525. APPENDIX Assumption 1(Smoothness).The nominal dynamics ¯M(χ), ¯C(χ,˙χ), and ¯f(χ,˙χ)are continuously differentiable. ¯M(χ) is assumed to be symmetric positive defini...

    2011