Pith. sign in

REVIEW 4 major objections 6 minor 36 references

Enhancing Robotic System Robustness via Lyapunov Exponent-Based Optimization

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read The signed sum of the Lyapunov exponents of a closed-loop robot is a robustness metric that differentiable simulation can minimize.

desk verdict A genuinely new differentiable Lyapunov-exponent objective for robot co-design, but the paper's scalar metric Lλ does not actually guarantee the non-divergence it is claimed to measure. read the letter →

arxiv 2412.06776 v1 pith:SZPZZKTI submitted 2024-12-09 cs.RO

G. Fadini , S. Coros This is my paper · ORCID
classification cs.RO
keywords Lyapunovexponentsrobustnessmetricdifferentiablesimulationco-designcontact-richdynamicslimitcyclesleggedlocomotiongradient-basedoptimization
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

Robustness to perturbations is hard to define, let alone optimize. The paper argues that a single number, the signed sum $L_\lambda = \mathbf{1}^\top \lambda$ of the Lyapunov exponents of the closed-loop system, captures how strongly nearby trajectories converge, and that making this number more negative improves the robot's ability to reject disturbances. The method computes $\lambda$ by differentiating a long simulation rollout: at each step, the Jacobian $\mathrm{d}\Phi(x_i)$ of the state transition is obtained from a differentiable physics simulator, the spectrum is extracted from an SVD of $\mathrm{d}\Phi^\top\mathrm{d}\Phi$, and the time average of the log singular values gives $\lambda$. Because the whole pipeline is differentiable, $L_\lambda$ can be minimized together with task costs by gradient-based co-optimization of hardware and policy parameters. This matters because, if valid, it turns the vague goal of robustness into a concrete objective usable for analysis, design, and control tuning in high-degree-of-freedom, contact-rich robots.

What carries the argument

The load-bearing object is the finite-horizon Lyapunov spectrum computed from a differentiable rollout. Let $\Phi(x_i)$ be one simulation step and $\mathrm{d}\Phi(x_i)$ its Jacobian; the paper estimates the exponents as $$ \$\lambda$ = \frac{1}{2N\$\Delta$ t}\sum_{i=1}^N \log(\Sigma_i), $$ where $\Sigma_i$ are the singular values of $\mathrm{d}\Phi(x_i)^\top\mathrm{d}\Phi(x_i)$ — an SVD variant chosen so that floating-base robots with non-square Jacobians remain tractable. The robustness score is the signed sum $L_\lambda = \mathbf{1}^\top\lambda$, which the paper identifies with the degree of hypervolume contraction in phase space: negative $L_\lambda$ means volumes shrink and trajectories converge. Differentiability of the rollout and of the SVD is what lets this scalar be optimized by first-order methods for hardware parameters and policy gains alike.

What would settle it

Run a sweep of gains and designs, rank them by $L_\lambda$, and physically perturb the best and worst: if any controller with more negative $L_\lambda$ rejects a finite push, terrain drop, or motor-torque noise worse than a less negative one, the proxy is falsified as a general robustness metric. A concrete version is to apply an impulsive side force to the optimized manipulator of Table I and measure settling time and success rate against a nominal design; the ranking by $L_\lambda$ predicts the optimized design wins every such test.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the signed sum of the Lyapunov exponents, $L_\lambda = \mathbf{1}^\top \lambda$, is an effective robustness metric for robotic systems, naturally covering fixed points and limit cycles, and that minimizing it via differentiable simulation improves perturbation rejection. Evidence is reported across three case studies: co-optimizing a two-link planar manipulator's link lengths, concentrated masses, and PD gains moves $L_\lambda$ from $-2.5\cdot 10^{-3}$ to $-1.3\cdot 10^{-2}$ and makes 100 randomly started trajectories converge more steadily; for a 16-joint spider falling from 1 m, the metric ranks gain combinations, with heavier damping and lower stiffness appearing more robust; and for a crawling quadruped with 8 actuated joints and a floating base, optimizing a 9-parameter sinusoidal policy yields $L_\lambda = -0.61$ versus $-0.56$ for the baseline. The paper presents this as the first gradient-based use of Lyapunov-exponent theory on a contact-rich, high-dimensional legged system.

Load-bearing premise

The approach treats the more-negative-is-more-robust relation as axiomatic: no proof, benchmark, or quantitative link to finite disturbances, actuator noise, or task performance is given, so optimizing $L_\lambda$ could simply produce overdamped or degenerate designs that shrink phase-space volume without rejecting the disturbances that matter.

Editorial extensions

If this is right

  • Co-design of robot morphology and controller can be posed as minimizing one scalar, $L_\lambda$, alongside task costs, with gradients obtained from a differentiable simulator.
  • The metric extends naturally to limit cycles, so periodic locomotion policies can be robustified without first stabilizing an equilibrium.
  • The same $L_\lambda$ value can be used to rank controllers or designs offline, as in the spider study where gain maps reveal a damping-versus-stiffness trade-off.
  • Because the full spectrum is computed rather than a single exponent, the method can in principle distinguish chaotic divergence in some directions from contraction in others.
  • Long-horizon rollouts used to estimate $\lambda$ may degrade gradient quality in contact-rich loco-manipulation, a limitation the paper explicitly acknowledges.

Reading between the lines

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

  • Editorial inference: because $L_\lambda$ rewards phase-space volume contraction only, unweighted minimization may favor overdamped, energy-dissipating designs; a testable extension is to include a task-performance weight or a finite-disturbance validation loop in the loss.
  • Editorial inference: the asymptotic average may not reflect finite-time robustness, so finite-time Lyapunov exponents or explicit disturbance injection would be a natural complement to this metric.
  • Editorial inference: for non-smooth contact dynamics the ergodic assumptions behind classical Lyapunov exponents may be questionable, making the quantity best read as a practical finite-horizon indicator rather than a rigorous global characteristic.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes a differentiable robustness metric for robotic systems, defined as the signed sum of Lyapunov exponents Lλ = 1^T λ, and uses it as an optimization objective in a differentiable-simulation framework. The method is demonstrated on three systems: a two-link planar manipulator whose link lengths, masses, and PD gains are co-optimized; a 16-joint spider robot whose joint PD gains are varied to assess falling robustness; and a quadruped whose periodic gait policy parameters are optimized. The central claim is that minimizing Lλ improves the ability of a robot to reject perturbations, with the paper reporting more negative Lλ values and qualitatively more convergent trajectories after optimization.

Significance. The paper addresses an important and open problem: obtaining a quantitative, computable robustness measure for high-degree-of-freedom, contact-rich robotic systems. The computational pipeline, which differentiates through long simulations to optimize a stability-related objective, is creative and potentially valuable for robot co-design. The range of test cases is a strength, and the volume-contraction interpretation of Lλ is mathematically meaningful. However, the central identification of Lλ with 'non-divergence of trajectories' is not valid in general, and the experimental validation is largely circular because the optimized quantity is used as its own evidence of success. With a corrected metric and independent finite-perturbation validation, the approach could be a solid contribution; as presented, the main claim is not yet supported.

major comments (4)
  1. [Section III, Eq. (9)] The paper defines robustness as 'trajectories which are not diverging one from the others' and selects Lλ = 1^T λ as the metric. This identification is not valid: the sum of Lyapunov exponents measures the average logarithmic rate of phase-space volume contraction, whereas non-divergence of nearby trajectories in every direction is governed by the largest exponent, λ_max. A negative sum can coexist with a positive largest exponent. For example, the linear map A1 = diag(1.05, 0.1) has Lλ = ln(1.05) + ln(0.1) ≈ -2.25 but is unstable along the first coordinate, while A2 = diag(0.8, 0.8) has Lλ ≈ -0.45 and contracts every direction. The proposed objective would rank A1 as more robust, contradicting the stated criterion. The authors should either replace Lλ with a metric that constrains λ_max < 0 or uses the full spectrum, or provide a concrete argument and experimental evidence that the robotic systems considered cannot exhibit this phenomenon.
  2. [Section II-D, Eq. (8) and Algorithm 1] The numerical procedure averages the log singular values of each local Jacobian dΦ(x_i) without reorthonormalizing a tangent basis between steps, so it does not compute the Oseledets Lyapunov spectrum. The individual components of Eq. (8) are not the Lyapunov exponents in general; only the sum of the components has a valid volume-contraction reading, because log det(∏ dΦ_i) = ∑ log det(dΦ_i). Consequently, the Van der Pol 'spectrum' in Fig. 2 cannot validate the individual exponents. The paper should either implement a standard QR/SVD algorithm with periodic reorthonormalization (e.g., the Benettin method) or explicitly restrict all claims to the summed quantity Lλ.
  3. [Section IV-B through IV-D] The experimental validation is insufficient because no independent, non-Lλ robustness measure is reported. The manipulator and quadruped results show that the optimized parameters improve Lλ, but since Lλ was the optimization objective, this is partly by construction; the random-start trajectory plots are qualitative and are not summarized by any quantitative disturbance-amplification or task-success metric. The quadruped change (Lλ from -0.56 to -0.61) is small and is not tied to any behavioral improvement. I recommend adding quantitative comparisons under finite perturbations (e.g., impulse disturbances, actuator noise, terrain variations), reporting quantities such as maximum deviation, settling time, or success rate, and comparing against a baseline optimized for the task alone.
  4. [Section IV-D and Section III-A] The theoretical framework assumes an autonomous transition map Φ(x_i), and Section III-A states that u_i = Π(x_i) makes the system autonomous. However, the quadruped policy in Eqs. (11)-(12) is explicitly time-dependent through the sinusoidal reference q_ref(t). A time-periodically forced system is not autonomous, and the standard Oseledets Lyapunov exponents are not directly applicable without augmenting the state with time (or using Floquet theory for periodic orbits). The paper should clarify how Eq. (1) applies to this experiment or use a time-augmented formulation.
minor comments (6)
  1. [Section II-D] The sentence 'As in floating base robotics, it may happen that dΦ_i is not a square matrix applying (8) is not viable' appears to refer to Eq. (7), not Eq. (8), and should be restructured for readability.
  2. [Section II-B] The notation |dΦ(x_i)| for a matrix is undefined; the volume expansion/contraction statement should be phrased in terms of the determinant or the singular values of the Jacobian.
  3. [Section IV-D] The phrase 'We compare the the results' contains a duplicated article.
  4. [Section V] The word 'Lypaunov exponents' in the conclusions is a typo; it should be 'Lyapunov exponents'.
  5. [Fig. 6 caption] The caption reads 'The two plot on the left' and should read 'The two plots on the left'.
  6. [Section III] The phrase 'where : 1 is the ones vector' contains a stray colon and should be 'where 1 is the ones vector'.

Circularity Check

1 steps flagged · score 6.0 of 10

Quantitative validation is partly by construction: robustness is defined as the scalar Lλ, the optimization objective includes Lλ, and the results report Lλ improvement as evidence of enhanced robustness; only qualitative trajectory plots provide independent support.

  1. self definitional [Section III, Eq. (9); validated in Section IV-B Tab. I and Section IV-D]
    "For robustness, we identify the system property to possess trajectories which are not diverging the one from the others, so, given these results, as a metric of robustness Lλ, the signed sum of the Lyapunov exponent components is taken: Lλ= 1⊕λ"

    The paper defines robustness as the scalar Lλ in Eq. (9), then includes Lλ in the differentiable cost function (Section III-A: 'including Lλ') and minimizes that cost. The results in Tab. I and Section IV-D then report the improved Lλ value as evidence of improved robustness: 'the metric of the robustness of the optimized robot Lλ is over 5 times higher than the nominal one' and 'the value of the optimized policy for robustness is Lλ=−0.61 while the baseline was Lλ=−0.56'. Successfully minimizing an objective term and then reporting that the same term improved is a tautology, not an independent validation of the metric. The random-start trajectory plots (Fig.

full rationale

I walked the paper's derivation chain from the sensitivity analysis (Eqs. 3-5) through the Lyapunov exponent definition (Eqs. 6-8) to the proposed robustness metric Lλ=1⊕λ (Eq. 9). The step from 'trajectories which are not diverging' to 'signed sum of the Lyapunov exponent components' is an identification, not a derivation; that is an assumption rather than a circularity by itself. The circularity enters in the experimental validation: Lλ is placed in the optimization loss, minimized by gradient descent, and then reported as the measured robustness improvement. Any successful optimization of a cost containing Lλ will lower Lλ, so the quantitative headline results (manipulator Lλ from -2.5e-3 to -1.3e-2; quadruped Lλ from -0.56 to -0.61) are forced by construction. The independent content is limited to qualitative trajectory plots and a parameter sweep whose robustness values are again Lλ. I found no load-bearing self-citation chain: the Fadini self-citations [2,3,8] support prior co-design work but are not used to justify Eq. (9). The separate mathematical concern that Lλ<0 can coexist with a positive largest Lyapunov exponent, so volume contraction does not imply non-divergence of nearby trajectories, is a correctness risk rather than a circularity and is therefore not part of this score. Because the central quantitative validation reduces to the definition of the metric while some independent qualitative evidence exists, the appropriate score is 6 (partial circularity).

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

The central claim rests on standard ergodic theory plus an unbenchmarked equivalence between volume contraction and robustness; the reported experimental conclusions depend on hand-picked horizons, under-reported optimized parameters, and a time-dependent policy that breaks the assumed autonomous structure.

free parameters (5)
  • simulation horizon and timestep (manipulator) = T = 2 s, dt = 1e-3 s
    Finite-horizon average is used as a proxy for the asymptotic Lyapunov exponent; no convergence or sensitivity study is reported.
  • simulation horizon and timestep (quadruped) = T = 10 s, dt = 1e-2 s
    Same proxy issue; the authors themselves note that long-horizon differentiable simulation can produce poor gradients [35].
  • manipulator co-optimization parameters = L=(0.9,0.1) m, m=(0.9,0.01) kg, Kd=(31.7,21.7), Kp=(59.8,59.8), Kc=(0.75,1.78)
    These values are optimized by Adam to minimize a cost containing Lλ; no cross-validation or sensitivity analysis is given, and the second link is nearly degenerate.
  • quadruped policy parameters = 9 parameters (amplitudes, frequencies, phases, delta, PD gains); final values not reported
    Policy parameters are optimized to improve Lλ, but the numeric results are omitted, preventing reproduction.
  • spider PD gain grid = normalized gains in Fig. 9; exact values not reported
    The parameter sweep that produces the Lλ trend is not described numerically.
assumptions (5)
  • standard math Oseledets multiplicative ergodic theorem guarantees existence of Lyapunov exponents for ergodic systems.
    Invoked in Section II-C as the theoretical foundation for the spectrum and the limit in Eq. (6).
  • domain assumption The closed-loop robotic system can be represented as an autonomous Markovian transition map Φ(x_i) with a fixed controller Π.
    Used in Section III-A, Eq. (10). It is violated by the quadruped policy in Eq. (11), which depends explicitly on time t unless time is augmented into the state.
  • domain assumption A finite-time average over N simulation steps approximates the infinite-horizon Lyapunov exponents.
    Used in Eq. (6)-(8) and Algorithm 1; no error bound, convergence criterion, or attractor reachability check is provided.
  • domain assumption The differentiable simulator provides accurate Jacobians dΦ_i and stable gradients through contacts.
    Required by the whole pipeline in Section IV-A; the paper cites known issues with contact-rich differentiable simulation and gradient quality [35].
  • ad hoc to paper The signed sum of Lyapunov exponents is a monotone proxy for robustness.
    Introduced in Section III, Eq. (9); not derived from first principles and not validated against external robustness metrics.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Enhancing Robotic System Robustness via Lyapunov Exponent-Based Optimization." pith.science (2026). https://pith.science/paper/SZPZZKTI

@misc{pith2026241206776,
  author       = {Pith},
  title        = {Pith review of: Enhancing Robotic System Robustness via Lyapunov Exponent-Based Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZPZZKTI}},
  note         = {Machine review of arXiv:2412.06776}
}
read the original abstract

We present a novel approach to quantifying and optimizing stability in robotic systems based on the Lyapunov exponents addressing an open challenge in the field of robot analysis, design, and optimization. Our method leverages differentiable simulation over extended time horizons. The proposed metric offers several properties, including a natural extension to limit cycles commonly encountered in robotics tasks and locomotion. We showcase, with an ad-hoc JAX gradient-based optimization framework, remarkable power, and flexi-bility in tackling the robustness challenge. The effectiveness of our approach is tested through diverse scenarios of varying complexity, encompassing high-degree-of-freedom systems and contact-rich environments. The positive outcomes across these cases highlight the potential of our method in enhancing system robustness.

Figures

Figures reproduced from arXiv: 2412.06776 by the authors.

Figure 1
Figure 1. The transition map Φ maps an uncertainty of radius 𝜀 around 𝑥𝑖 into an ellipsoid 𝜀 ⊤dΦ⊤(𝑥𝑖)dΦ(𝑥𝑖)𝜀. the next one. This dependency is written as: 𝑥𝑖+1 = Φ(𝑥𝑖) (1) Intuitively Φ can be interpreted as a single forward simu￾lation step which links one state 𝑥𝑖 to the next one 𝑥𝑖+1. In a complete simulation rollout, the state transition map is applied iteratively for 𝑛 steps starting from the initial condition 𝑥0. This l… view at source ↗
Figure 3
Figure 3. Parallelized Lyapunov spectrum computation: first a [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Robot manipulator scheme TABLE I: Comparison of nominal and optimized parameters Parameter Nominal Optimized Lengths 𝐿1, 𝐿2 [m] (1.0, 1.0) (0.9, 0.1) Masses 𝑚1, 𝑚2 [kg] (0.5, 0.5) (0.9, 0.01) Damping 𝐾𝑑 [Ns/rad] (5.0, 5.0) (31.7, 21.7) Stiffness 𝐾𝑝 [Nm/rad] (20.0, 20.0) (59.8, 59.8) Stiffness 𝐾𝑐 [Nm/rad] (1.0, 1.0) (0.75, 1.78) Cost 1.06 0.21 Robustness L𝜆 −2.5 · 10−3 −1.3 · 10−2 representation which could be differ… view at source ↗
Figures from the paper (4 more)
Figure 8
Figure 8. Figure 8: Model of the 16 DoF spider 8a and 100 simulation [PITH_FULL_IMAGE:figures/full_fig_p005_8.png]
Figure 7
Figure 7. Figure 7: Cost trend during the co-optimization of the manip [PITH_FULL_IMAGE:figures/full_fig_p005_7.png]
Figure 10
Figure 10. Figure 10: Quadruped robot the parameter space, across the joints, we implement shared amplitudes 𝐴0, 𝐴1 and frequencies 𝐹0, 𝐹1 and phases 𝑃0, 𝑃1 for the different hip and knee joints. An additional parameter 𝛿 adds a phase between each pair of opposing legs. 𝑞{FR, RL} HFE = 𝑞re…
Figure 11
Figure 11. Figure 11: Periodic trajectory obtained in simulation with the [PITH_FULL_IMAGE:figures/full_fig_p006_11.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

36 extracted references · 32 canonical work pages

  1. [1]

    What Is Morphological Computation? On How the Body Contributes to Cognition and Control

    Vincent C. M ¨uller and Matej Hoffmann. “What Is Morphological Computation? On How the Body Contributes to Cognition and Control”. In: Artificial Life 23.1 (Feb. 2017), pp. 1–24

    work page 2017

  2. [2]

    Computational design of energy-efficient legged robots: Optimizing for size and actuators

    G. Fadini et al. “Computational design of energy-efficient legged robots: Optimizing for size and actuators”. In: 2021 IEEE Inter- national Conference on Robotics and Automation (ICRA) . 2021 IEEE International Conference on Robotics and Automation (ICRA). IEEE

    work page 2021

  3. [3]

    Co-designing versatile quadruped robots for dy- namic and energy-efficient motions

    G. Fadini et al. “Co-designing versatile quadruped robots for dy- namic and energy-efficient motions”. working paper or preprint. July 2023

    work page 2023

  4. [4]

    Computational co-optimization of design param- eters and motion trajectories for robotic systems

    Sehoon Ha et al. “Computational co-optimization of design param- eters and motion trajectories for robotic systems”. In: The Interna- tional Journal of Robotics Research 37.13 (Dec. 2018), pp. 1521–

    work page 2018

  5. [5]

    Skaterbots: optimization-based design and motion synthesis for robotic creatures with legs and wheels

    Moritz Geilinger et al. “Skaterbots: optimization-based design and motion synthesis for robotic creatures with legs and wheels”. In: ACM Transactions on Graphics 37.4 (Aug. 10, 2018), pp. 1–12

    work page 2018

  6. [6]

    Robust Co-Design: Coupling Mor- phology and Feedback Design Through Stochastic Programming

    Gabriel Bravo-Palacios et al. “Robust Co-Design: Coupling Mor- phology and Feedback Design Through Stochastic Programming”. In: Journal of Dynamic Systems, Measurement, and Control (Feb. 1, 2022)

    work page 2022

  7. [7]

    Large-Scale ADMM-based Co-Design of Legged Robots

    Gabriel Bravo-Palacios and Patrick M. Wensing. “Large-Scale ADMM-based Co-Design of Legged Robots”. In: IEEE/RSJ Int. Conf. on Intelligent Robots and Systems . 2022

    work page 2022

  8. [8]

    Simulation aided co-design for robust robot op- timization

    G. Fadini et al. “Simulation aided co-design for robust robot op- timization”. In: IEEE Robotics and Automation Letters 7.4 (Oct. 2022)

    work page 2022

Show all 36 references
  1. [9]

    Co-optimization of Acrobot Design and Controller for Increased Certifiable Stability

    Lasse Jenning Maywald et al. “Co-optimization of Acrobot Design and Controller for Increased Certifiable Stability”. In: IEEE Int. Conf. on Intelligent Robots and Systems . 2022

    2022

  2. [10]

    Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability

    F. Girlanda et al. “Robust Co-Design of Canonical Underactuated Systems for Increased Certifiable Stability”. In: IEEE ICRA. 2024

    2024

  3. [11]

    Trajectory Generation for Minimum Closed-Loop State Sensitivity

    Paolo Robuffo Giordano, Quentin Delamare, and Antonio Franchi. “Trajectory Generation for Minimum Closed-Loop State Sensitivity”. In: IEEE Int. Conf. on Robotics & Automation . Brisbane, Australia, 2018

    2018

  4. [12]

    ADD: Analytically Differentiable Dynamics for Multi-Body Systems with Frictional Contact

    Moritz Geilinger et al. ADD: Analytically Differentiable Dynamics for Multi-Body Systems with Frictional Contact. July 2, 2020. arXiv: 2007.00987[cs]

    2020 arXiv

  5. [13]

    H. J. Terry Suh et al. Do Differentiable Simulators Give Better Policy Gradients? Aug. 22, 2022. arXiv: 2202.00817[cs]. URL: http: //arxiv.org/abs/2202.00817

    2022 arXiv

  6. [14]

    Lyapunov exponents for non-smooth dynamical sys- tems

    Markus Kunze. “Lyapunov exponents for non-smooth dynamical sys- tems”. In: Non-Smooth Dynamical Systems . Ed. by Markus Kunze. Berlin, Heidelberg: Springer Berlin Heidelberg, 2000, pp. 63–140

    2000

  7. [15]

    Lyapunov Exponents

    Lu ´ıs Barreira. Lyapunov Exponents . Cham: Springer International Publishing, 2017. DOI: 10.1007/978-3-319-71261-1

    2017 doi

  8. [16]

    Applications of Chaotic Dynamics in Robotics

    Xizhe Zang et al. “Applications of Chaotic Dynamics in Robotics”. In: International Journal of Advanced Robotic Systems 13.2 (Mar. 1, 2016), p. 60. DOI: 10.5772/62796

    2016 doi

  9. [17]

    On stabilization of bipedal robots during dis- turbed standing using the concept of Lyapunov exponents

    C. Yang and Q. Wu. “On stabilization of bipedal robots during dis- turbed standing using the concept of Lyapunov exponents”. In: 2006 American Control Conference . 2006 American Control Conference. Minneapolis, MN, USA: IEEE, 2006, 6 pp

    2006

  10. [18]

    Stability Analysis of Bipedal Robots Using the Concept of Lyapunov Exponents

    Liu Yunping et al. “Stability Analysis of Bipedal Robots Using the Concept of Lyapunov Exponents”. In: Mathematical Problems in Engineering 2013 (2013), pp. 1–4

    2013

  11. [19]

    Arnold and V

    L. Arnold and V . Wishtutz. Lyapunov Exponents. V ol. 1186. Lecture Notes in Math. Bremen, 1984. Springer, 1986

    1984

  12. [20]

    Ergodic theory of chaos and strange attractors

    J.-P. Eckmann and D. Ruelle. “Ergodic theory of chaos and strange attractors”. In: Rev. Mod. Phys. 57 (1985), p. 617

    1985

  13. [21]

    Characterizing dynamics with covariant Lyapunov vectors

    F. Ginelli et al. “Characterizing dynamics with covariant Lyapunov vectors”. In: Phys. Rev. Lett. 99 (2007), p. 130601

    2007

  14. [22]

    Characterization of strange attrac- tors

    P. Grassberger and I. Procaccia. “Characterization of strange attrac- tors”. In: Phys. Rev. Lett. 50 (1983), p. 346

    1983

  15. [23]

    What are Lyapunov exponents, and why are they interesting?

    Amie Wilkinson. “What are Lyapunov exponents, and why are they interesting?” In: Bulletin of the American Mathematical Society 54.1 (Sept. 6, 2016), pp. 79–105

    2016

  16. [24]

    Nonlinear Dynamics: A Concise Introduction Interlaced with Code

    George Datseris and Ulrich Parlitz. Nonlinear Dynamics: A Concise Introduction Interlaced with Code . Undergraduate Lecture Notes in Physics. Cham: Springer International Publishing, 2022

    2022

  17. [25]

    A multiplicative ergodic theorem. Lyapunov char- acteristic numbers for dynamical systems

    V . I. Oseledets. “A multiplicative ergodic theorem. Lyapunov char- acteristic numbers for dynamical systems”. In: Transactions of the Moscow Mathematical Society 19 (1968). Translated from Russian, pp. 197–231

    1968

  18. [26]

    DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics

    George Datseris. “DynamicalSystems.jl: A Julia software library for chaos and nonlinear dynamics”. In: Journal of Open Source Software 3.23 (Mar. 2018), p. 598. DOI: 10.21105/joss.00598

    2018 doi

  19. [27]

    Automatic Differentiation for Complex Valued SVD

    Shi-Xin Zhang et al. “Automatic Differentiation for Complex Valued SVD”. In: arXiv: Numerical Analysis (2019). arXiv: 1909.02659 [math.NA]

    2019 arXiv

  20. [28]

    A theory of the amplitude of free and forced triode vibrations

    B. van der Pol. “A theory of the amplitude of free and forced triode vibrations”. In: Radio Review 1 (1920), pp. 701–710, 754–762

    1920

  21. [29]

    Liouville’s theorem

    Robert Phillips. “Liouville’s theorem”. In: Pacific Journal of Mathe- matics 28.2 (1969). Discusses Liouville’s theorem in the context of conformal mappings, pp. 397–405

    1969

  22. [30]

    Mujoco: A physics engine for model-based control

    Emanuel Todorov, Tom Erez, and Yuval Tassa. “Mujoco: A physics engine for model-based control”. In: 2012 IEEE/RSJ Interna- tional Conference on Intelligent Robots and Systems . IEEE. 2012, pp. 5026–5033

    2012

  23. [31]

    JAX: composable transformations of Python+NumPy programs

    James Bradbury et al. JAX: composable transformations of Python+NumPy programs. https://github.com/google/ jax. Version 0.3.13. 2018

    2018

  24. [32]

    Impedance Control: An Approach to Manipulation: Part I—Theory

    Neville Hogan. “Impedance Control: An Approach to Manipulation: Part I—Theory”. In: Journal of Dynamic Systems, Measurement, and Control 107.1 (Mar. 1, 1985), pp. 1–7

    1985

  25. [33]

    Adam: A Method for Stochastic Optimization

    Diederik P. Kingma and Jimmy Ba. “Adam: A Method for Stochastic Optimization”. In: CoRR abs/1412.6980 (2014)

    2014 arXiv

  26. [34]

    The DeepMind JAX Ecosystem

    DeepMind et al. The DeepMind JAX Ecosystem . 2020

    2020

  27. [35]

    Rethinking Optimization with Differentiable Simulation from a Global Perspective

    Rika Antonova et al. Rethinking Optimization with Differentiable Simulation from a Global Perspective. June 28, 2022. arXiv: 2207. 00167[cs,stat]

    2022

  28. [1536]

    ISSN : 0278-3649, 1741-3176

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.