Pith. sign in

REVIEW 3 major objections 5 minor 1 cited by

Time-Varying Directional State Transition Tensor for Orbit Uncertainty Propagation

T0 review · 3 major / 5 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 A time-varying directional state transition tensor predicts orbital uncertainty at any epoch along the orbit with nearly the accuracy of the full state transition tensor, while cutting computational cost by roughly 94% at third order.

desk verdict Real extension of DSTT to time-varying sensitive directions, with an eigenvalue-crossing caveat the authors acknowledge but don't solve. read the letter →

arxiv 2412.07060 v1 pith:3CPEAVKM submitted 2024-12-10 astro-ph.IM

Xingyu Zhou , Roberto Armellin , Dong Qiao , Xiangyu Li This is my paper · ORCID
classification astro-ph.IM
keywords orbituncertaintypropagationstatetransitiontensorCauchy-Greeneigenvectorderivativesthree-bodyproblemnear-rectilinearhalotemporarycapturedirectionalsensitivity
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

Orbital uncertainty propagation normally requires either a full state transition tensor, whose cost grows exponentially with order, or a directional variant that is valid only at one pre-chosen epoch. This paper proposes a time-varying directional state transition tensor (TDSTT) that integrates the sensitive directions (eigenvectors of the Cauchy-Green tensor) together with the high-order Taylor coefficients, so the same run yields uncertainty predictions at any epoch along the orbit. On a temporary-capture orbit in the Sun-Jupiter system and on a near-rectilinear halo orbit in the Earth-Moon system, the TDSTT is claimed to match the accuracy of the full STT and the DSTT while being about 94% faster than the full STT at third order and hundreds of times faster than the DSTT for evolution analyses. If true, this makes high-order nonlinear uncertainty propagation practical for conjunction assessment and onboard applications where the time history matters.

What carries the argument

The central object is the time-varying directional state transition tensor, built from the eigenvectors of the Cauchy-Green tensor $C=\Phi^T\Phi$ associated with the largest eigenvalues and integrated along the orbit. The workhorse identities are the eigenvector derivative formulas: Nelson's method computes $\dot{\xi}_k$ using only the pair $(\lambda_k,\xi_k)$, and the linear representation $\dot{\xi}_k = B_{k,p}\xi_p$ turns the time derivative of the projection matrix into a few extra terms. Orthogonality $R_{\gamma,k}R_{p,k} = \delta_{\gamma p}$ collapses those terms, reducing the per-step complexity for the second-order TDSTT from $O(n+ n^2 + 2 n^4 m^2)$ to $O(n+n^2+2m)$.

What would settle it

Propagate a second-order TDSTT with two sensitive directions through the Jupiter temporary-capture orbit, and at epochs after the second and third eigenvalues cross, compare the predicted covariance with a full Monte Carlo or full STT; if the position mean absolute error ratio to the full STT jumps by more than an order of magnitude beyond the roughly $10^{-6}$ level reported at the final epoch, the fixed-label assumption is violated.

Watch

Extended reading notes

Core claim

The central claim is that the directional state transition tensor can be made time-dependent without losing accuracy. The authors derive differential equations for the derivatives of the Cauchy-Green tensor's eigenvalues and eigenvectors (using Nelson's method for the eigenvectors), then use orthogonality of the sensitive directions to simplify the TDSTT derivative equations. Because the eigenvector directions are integrated forward with the nominal trajectory, the resulting tensors are valid at every output epoch, not only the final epoch. Numerical experiments report mean absolute errors matching the full STT and DSTT to within a few percent, while the eigenvalue along the most sensitive direction reaches roughly $10^{12}$ at the final epoch of the Sun-Jupiter case, a regime where linear propagation fails.

Load-bearing premise

The method assumes that the direction it labels as most sensitive at the start keeps being the most important direction all the way; if two directions swap importance mid-flight, the method may keep tracking a direction that is no longer dominant.

Editorial extensions

If this is right

  • Uncertainty statistics can be extracted at any epoch from a single TDSTT integration, so time histories for conjunction assessment no longer require a separate DSTT run per epoch.
  • At third order with two sensitive directions, the TDSTT is about 94% faster than the full STT in the two tested systems, and the number of integrated variables grows polynomially rather than exponentially with order.
  • For evolution analyses over many epochs, the TDSTT is hundreds of times faster than the direct-way DSTT, which must be re-integrated for each predefined epoch.
  • The accuracy loss is small: in the Jupiter case, third-order TDSTT errors are within a few percent of the full STT, while the first-order STM has mean relative errors above 7.5%.
  • Because the TDSTT reuses the same integrated tensor at every output epoch, it can replace repeated Monte Carlo runs in onboard covariance prediction when a dense uncertainty timeline is required.

Reading between the lines

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

  • If the eigenvector labeling persists, the TDSTT could be combined with sequential orbit determination: a single integration would give the covariance history needed for filtering without repeated tensor integrations.
  • The observed eigenvalue swap suggests a practical extension: monitor the order of propagated eigenvalues and re-sort the sensitive directions when two eigenvalues cross, at the cost of a discontinuity in the tensor ODE.
  • Because the method tracks the leading Cauchy-Green eigendirections, it could serve as a cheap detector of dynamical sensitivity transitions, such as gravitational keyholes or capture boundaries, flagging epochs where the dominant error direction changes.
  • The method's accuracy at any epoch could enable onboard collision-risk screening on small computers, where a polynomial-cost tensor integration is far more feasible than repeated full-tensor runs.
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

3 major / 5 minor

Summary. The paper proposes a time-varying directional state transition tensor (TDSTT) for orbital uncertainty propagation. The TDSTT extends the DSTT by integrating the eigenvalues and eigenvectors of the Cauchy-Green tensor alongside the nominal trajectory, the state transition matrix, and the high-order TDSTT terms, so that the sensitive directions are no longer frozen at a predefined epoch. The authors derive the derivatives of the eigenvalues and eigenvectors, use a projection approximation to obtain simplified ODEs for second- and higher-order TDSTTs, and present a complexity analysis. The method is tested on a temporary-capture orbit in the Sun-Jupiter CRTBP and on a 9:2 near-rectilinear halo orbit in the Earth-Moon CRTBP, with Monte Carlo validation against the full STT and the original DSTT. The reported results show accuracy close to that of the full STT and large computational savings, especially for epoch-by-epoch evolution analyses.

Significance. If the method performs as claimed, it is a practically useful reduced-order uncertainty propagation tool for highly nonlinear three-body dynamics, combining the accuracy of high-order STTs with a much smaller number of integrated variables and enabling analysis at arbitrary epochs without re-integrating the DSTT. The paper's strengths include validation against independent Monte Carlo and full-STT references, no fitted parameters, a public code link, and a clear statement of the main algorithmic steps. The central idea is original relative to the constant-direction DSTT of Boone and McMahon. However, the paper's general claims of 'any epoch' capability and 'nearly the same level of accuracy' are not fully supported because the method has no mechanism to handle eigenvalue crossings or label swaps, and because the derivation relies on a projection approximation for which no error bound is provided. These issues affect the robustness of the method in regimes that the presented test cases do not exercise.

major comments (3)
  1. [Sec. V.A, Fig. 2 and surrounding discussion] The manuscript explicitly states that the TDSTT cannot follow any swap in the magnitude of the eigenvalues. Since the eigenvector labels are fixed at the warm-start epoch and the eigenvector derivative formulas in Eqs. (38)-(40) require distinct eigenvalues, a crossing involving the largest eigenvalue, or a selected direction dropping out of the top-m set, would cause the TDSTT to keep aligning high-order terms along a direction that is no longer dominant. Near such a crossing, the coefficients B_{k,p} in Eq. (40) diverge as λ_p - λ_k tends to zero, making the ODEs stiff or singular. The two test cases are benign only because λ1 remains dominant and distinct; no numerical example exercises the crossing regime. The authors should either add a reordering/relabeling mechanism and test it on a case where the crossing involves the dominant direction, or explicitly restrict the 'any epoch' and 'nearly the same accuracy' claims and provide evidence of graceful degradation near crossings.
  2. [Sec. IV.B, Eqs. (46)-(48)] The derivation of the TDSTT ODEs replaces the full higher-order STTs by their projection onto the sensitive subspace defined by the rows of R. This is an approximation, as the paper acknowledges in Sec. V.A, but no error bound or quantitative validity condition is given. The projected expressions are then differentiated to produce Eqs. (56), (62), and (63), so the integrated TDSTT is not guaranteed to track the full STT. The numerical evidence shows good agreement in two examples, but it does not establish when the approximation is safe, especially when more than one direction is used or when eigenvalues are close. Please provide an error estimate in terms of the neglected eigenvalues or the spectral gap, or a numerical study that varies m and the crossing structure to delineate the regime in which the TDSTT can be trusted.
  3. [Sec. IV.D and Conclusion] The claim that the algorithm complexity grows polynomially with the expansion order is only true for m=1. The variable counts in Table 3 and the text's own formula n+n^2+(n+1)m+n(m^2+m^3) show that for m>1 the p-th order TDSTT contributes n m^p terms, which grows exponentially in the order P. The concluding sentence 'the algorithm complexity of the TDSTT grows polynomially with the order' should therefore be qualified to the m=1 case, or the complexity analysis should be revised to state the exponential-in-P scaling for general m.
minor comments (5)
  1. [Eq. (39)] The linear representation is written as dξ_k/dt = B_{k,p} ξ_k, but Eq. (38) shows that the derivative is a linear combination of the other eigenvectors, so the expression should be dξ_k/dt = B_{k,p} ξ_p with summation over p≠k. This typo is confusing and should be corrected, especially because Eq. (53) subsequently uses the correct form.
  2. [Eq. (19)] The fourth-order DSTT ODE contains inconsistent index expressions, including φ_{α,p1p2p3γ4} and φ_{α,p1d}; these should be corrected to the appropriate p4 and p1p4 index structures to match the other terms.
  3. [Table 3] The shape labels for the TDSTT rows appear to read R^{m×n×n} and R^{m×n×n×n}, but the variable counts n m^2 and n m^3 given in the text and in the Total row imply the shapes should be R^{n×m×m} and R^{n×m×m×m}.
  4. [General] The manuscript repeatedly uses 'identified matrix' where 'identity matrix' is meant (e.g., Sections III.B.2 and IV.C), and 'statical moments' should be 'statistical moments'.
  5. [Fig. 2] In Fig. 2, the eigenvalue swap after t≈0.5 nd is described in the text and is important, but the legend only shows λ̃1, λ̃2, λ̃3; adding a note or marker indicating the ordering ambiguity would help readers interpret the plot.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the TDSTT derivation is self-contained and its accuracy claims are benchmarked against independent Monte Carlo and full-STT references.

full rationale

The TDSTT derivation chain is self-contained rather than circular. The TDSTT coefficients are defined as projections of the full STT onto the time-varying eigenvectors of the Cauchy-Green tensor (Eq. 11), and the new differential equations (56), (62), and (63) follow from the exact product rule combined with an explicit low-rank closure (Eqs. 46-48), not from the target accuracy being inserted as an input. No parameter is fitted to reproduce the reported error levels; the accuracy claims are assessed against independent Monte Carlo sampling and full STT propagation (Figs. 6-9 and Tables 6, 11), which are external references rather than outputs of the method itself. The eigenvalue/eigenvector derivative formulas in Section III are standard mathematics (Nelson's method, cited to Ref. 45) with a clearly stated distinct-eigenvalue assumption, and the warm-start procedure is a numerical initialization, not a hidden fit. The paper itself flags a genuine limitation in Section V.A, stating that 'the TDSTT cannot follow any swap in the magnitude of the eigenvalues' and exhibiting a lambda_2/lambda_3 swap in Fig. 2; this is a robustness caveat about eigenvalue crossing, not circular reasoning, and it does not make the validation benchmarks self-referential. Self-citations (e.g., Refs. 3, 39, 44) provide context or prior applications and are not load-bearing for the new derivation. The consistency check that the propagated lambda_1 matches the CGT-computed lambda_1 (Fig. 3) is a numerical verification of the derived ODE, not a prediction obtained from the quantity being predicted. Overall, the central derivation and validation are independent of the paper's own conclusions, so no circular step rises above score 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The method introduces no fitted physical parameters. It relies on standard linear algebra for symmetric matrices, a projection approximation for high-order STT terms, and a domain assumption that CRTBP is the relevant dynamics. No new physical entities are postulated.

free parameters (3)
  • m (number of sensitive directions) = 1 or 2
    User-selected; accuracy and computational cost depend on it. Not fit to data.
  • P (Taylor expansion order) = 2 or 3
    User-selected; higher order improves accuracy at higher cost.
  • Warm-start epoch t' as a fraction of t_f = t_f / 100000 (Jupiter case)
    Chosen to avoid repeated CGT eigenvalues at t0; the value affects eigenvector initialization and computational cost.
assumptions (4)
  • domain assumption CGT eigenvalues are distinct along the integration interval
    Eigenvector derivative formulas in Eqs. (33), (38), and (45) divide by eigenvalue differences; repeated eigenvalues cause singularities. The paper notes this in Section III.B.
  • ad hoc to paper The selected eigenvectors, propagated continuously from warm start, remain the dominant sensitive directions for the whole interval
    TDSTT integrates fixed eigenvector labels; eigenvalue magnitude swaps (seen in the Jupiter case) mean tracked directions may no longer be the top-m. Section V.A discusses this potential limitation.
  • ad hoc to paper Full higher-order STTs can be approximated by their projection onto the sensitive subspace in Eqs. (46)-(48)
    Used to close the TDSTT ODEs; no error bound is provided.
  • domain assumption CRTBP is a sufficient dynamics model for the two test cases
    Numerical validation uses Sun-Jupiter and Earth-Moon CRTBP, not full ephemeris or higher-fidelity perturbations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Time-Varying Directional State Transition Tensor for Orbit Uncertainty Propagation." pith.science (2026). https://pith.science/paper/3CPEAVKM

@misc{pith2026241207060,
  author       = {Pith},
  title        = {Pith review of: Time-Varying Directional State Transition Tensor for Orbit Uncertainty Propagation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CPEAVKM}},
  note         = {Machine review of arXiv:2412.07060}
}
read the original abstract

The directional state transition tensor (DSTT) reduces the complexity of state transition tensor (STT) by aligning the STT terms in sensitive directions only, which provides comparable accuracy in orbital uncertainty propagation. The DSTT assumes the sensitive directions to be constant during the integration and only works at a predefined epoch. This paper proposes a time-varying STT (TDSTT) to improve the DSTT. The proposed TDSTT computes the sensitive directions with time; thereby, it can perform uncertainty propagation analysis at any point instead of only a predefined epoch as the DSTT does. First, the derivatives of the sensitive directions are derived. Then, the differential equations for the high-order TDSTTs are derived and simplified using the orthogonality of sensitive directions. Next, complexity analysis is implemented to show the advantages of the proposed TDSTT over the STT. Finally, the TDSTT is applied to solve orbital uncertainty propagation problems in highly nonlinear three-body systems. Numerical results show that the proposed TDSTT can yield nearly the same level of accuracy as the STT and DSTT. It is approximately 94% faster than the STT and has hundreds of improvements in speed over the DSTT when one wants to investigate the evolutions of orbital uncertainties.

Figures

Figures reproduced from arXiv: 2412.07060 by the authors.

Figure 4
Figure 4. One can see from Table 5 and Fig. 4 that the eigenvectors change as time evolves, and that’s why the DSTT is [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Improved Directional State Transition Tensors for Accurate Aerocapture Performance Analysis

    eess.SY 2025-12 conditional novelty 6.0 of 10

    State transition tensors are extended to aerocapture with reduced-dimension 'directional' versions whose reduction directions come from higher-order Cauchy-Green tensor eigenpairs, improving accuracy for energy and se...

Reference graph

Works this paper leans on

47 extracted references · 28 canonical work pages · cited by 1 Pith paper

  1. [1]

    A review of uncertainty propagation in orbital mechanics,

    Luo, Y., and Yang, Z., “A review of uncertainty propagation in orbital mechanics,”Progress in Aerospace Sciences, Vol. 89, 2017, pp. 23–39. https://doi.org/10.1016/j.paerosci.2016.12.002

    work page doi:10.1016/j.paerosci.2016.12.002 2017

  2. [2]

    Reviewofspacerelativenavigationbasedonangles-onlymeasurements,

    Gong,B.,Wang,S.,Li,S.,andLi,X.,“Reviewofspacerelativenavigationbasedonangles-onlymeasurements,” Astrodynamics, Vol. 7, No. 2, 2023, pp. 131–152. https://doi.org/10.1007/s42064-022-0152-2

    work page doi:10.1007/s42064-022-0152-2 2023

  3. [4]

    Perturbed initial orbit determination,

    Fossà, A., Losacco, M., and Armellin, R., “Perturbed initial orbit determination,”Astrodynamics, Vol. 8, 2024, pp. 401–416. https://doi.org/10.1007/s42064-023-0179-z. 33

    doi:10.1007/s42064-023-0179-z 2024

  4. [5]

    SemianalyticalMeasuresofNonlinearityBasedonTensorEigenpairs,

    Jenson, E.L., andScheeres, D.J., “SemianalyticalMeasuresofNonlinearityBasedonTensorEigenpairs,” Journal of Guidance, Control, and Dynamics, Vol. 46, No. 4, 2022, pp. 638–653. https://doi.org/10.2514/1.G006760

    doi:10.2514/1.g006760 2022

  5. [6]

    Bounding nonlinear stretching about spacecraft trajectories using tensor eigenpairs,

    Jenson, E. L., and Scheeres, D. J., “Bounding nonlinear stretching about spacecraft trajectories using tensor eigenpairs,”Acta Astronautica, Vol. 214, 2024, pp. 159–166. https://doi.org/10.1016/j.actaastro.2023.10.013

    doi:10.1016/j.actaastro.2023.10.013 2024

  6. [7]

    Collision avoidance maneuver optimization with a multiple-impulse convex formulation,

    Armellin, R., “Collision avoidance maneuver optimization with a multiple-impulse convex formulation,”Acta Astronautica, Vol. 186, 2021, pp. 347–362. https://doi.org/10.1016/j.actaastro.2021.05.046

    work page doi:10.1016/j.actaastro.2021.05.046 2021

  7. [8]

    Spacecraft collision avoidance challenge: Design and results of a machine learning competition,

    Uriot, T., Izzo, D., Simões, L. F., Abay, R., Einecke, N., Rebhan, S., Martinez-Heras, J., Letizia, F., Siminski, J., and Merz, K., “Spacecraft collision avoidance challenge: Design and results of a machine learning competition,”Astrodynamics, Vol. 6, No. 2, 2022, pp. 121–140. https://doi.org/10.1007/s42064-021-0101-5

    work page doi:10.1007/s42064-021-0101-5 2022

  8. [9]

    Nonlinear semi-analytical uncertainty propagation for conjunction analysis,

    Khatri, Y., and Scheeres, D. J., “Nonlinear semi-analytical uncertainty propagation for conjunction analysis,”Acta Astronautica, Vol. 203, 2023, pp. 568–576. https://doi.org/10.1016/j.actaastro.2022.12.009

    work page doi:10.1016/j.actaastro.2022.12.009 2023

Show all 47 references
  1. [10]

    Orbital Guidance Using Higher-Order State Transition Tensors,

    Boone, S., and McMahon, J., “Orbital Guidance Using Higher-Order State Transition Tensors,”Journal of Guidance, Control, and Dynamics, Vol. 44, No. 3, 2020, pp. 493–504. https://doi.org/10.2514/1.G005493

    2020 doi

  2. [11]

    Multi-Objective Optimization of Covariance and Energy for Asteroid Transfers,

    Jenson, E. L., and Scheeres, D. J., “Multi-Objective Optimization of Covariance and Energy for Asteroid Transfers,”Journal of Guidance, Control, and Dynamics, Vol. 44, No. 7, 2021, pp. 1253–1265. https://doi.org/10.2514/1.G005609

    2021 doi

  3. [12]

    Robust Space Trajectory Design Using Belief Optimal Control,

    Greco, C., Campagnola, S., and Vasile, M., “Robust Space Trajectory Design Using Belief Optimal Control,”Journal of Guidance, Control, and Dynamics, Vol. 45, No. 6, 2022, pp. 1060–1077. https://doi.org/10.2514/1.G005704

    2022 doi

  4. [13]

    Review of Trajectory Design and Optimization for Jovian System Exploration,

    Yang, H., Hu, J., Bai, X., and Li, S., “Review of Trajectory Design and Optimization for Jovian System Exploration,”Space: Science & Technology, Vol. 3, 2023, p. 36. https://doi.org/10.34133/space.0036

    2023 doi

  5. [14]

    Maneuvering Spacecraft Orbit Determination Using Polynomial Representation,

    Zhou, X., Qin, T., and Meng, L., “Maneuvering Spacecraft Orbit Determination Using Polynomial Representation,”Aerospace, Vol. 9, No. 5, 2022, p. 257. https://doi.org/10.3390/aerospace9050257

    2022 doi

  6. [15]

    Segmented Reconstruction of Low-Acceleration Orbital Maneuvers,

    Jia-Richards, O., “Segmented Reconstruction of Low-Acceleration Orbital Maneuvers,”Journal of Guidance, Control, and Dynamics, Vol. 1, No. 1, 2023, pp. 1–10. https://doi.org/10.2514/1.G007548

    2023 doi

  7. [16]

    Detection and estimation of spacecraft maneuvers for catalog maintenance,

    Pirovano, L., and Armellin, R., “Detection and estimation of spacecraft maneuvers for catalog maintenance,”Acta Astronautica, Vol. 215, 2024, pp. 387–397. https://doi.org/10.1016/j.actaastro.2023.12.016

    2024 doi

  8. [17]

    Orbit Determination and Thrust Estimation for Noncooperative Target Using Angle-Only Measurement,

    Zhang, Z., Shu, L., Zhang, K., Zhu, Z., Zhou, M., Wang, X., and Yin, W., “Orbit Determination and Thrust Estimation for Noncooperative Target Using Angle-Only Measurement,”Space: Science & Technology , Vol. 3, 2024, p. 73. https: //doi.org/10.34133/space.0073

    2024 doi

  9. [18]

    Autonomous navigation of an asteroid orbiter enhanced by a beacon satellite in a high-altitude orbit,

    Yin, W., Shi, Y., Shu, L., and Gao, Y., “Autonomous navigation of an asteroid orbiter enhanced by a beacon satellite in a high-altitude orbit,”Astrodynamics, Vol. 8, 2024, pp. 375–400. https://doi.org/10.1007/s42064-023-0172-6. 34

    2024 doi

  10. [19]

    Feasible domain analysis of heliocentric gravitational-wave detection configuration using semi-analytical uncertainty propagation,

    Qiao, D., Zhou, X., and Li, X., “Feasible domain analysis of heliocentric gravitational-wave detection configuration using semi-analytical uncertainty propagation,”Advances in Space Research, Vol. 72, 2023, pp. 4115–4131. https://doi.org/10.1016/j. asr.2023.08.011

    2023 doi

  11. [20]

    Direct Statistical Analysis of Nonlinear Systems: CADET,

    Gelb, A., and Warren, R. S., “Direct Statistical Analysis of Nonlinear Systems: CADET,”AIAA Journal, Vol. 11, No. 5, 1973, pp. 689–694. https://doi.org/10.2514/3.50507

    1973 doi

  12. [21]

    Analytical configuration uncertainty propagation of geocentric interferometric detection constellation,

    Qiao, D., Zhou, X., and Li, X., “Analytical configuration uncertainty propagation of geocentric interferometric detection constellation,” Astrodynamics, Vol. 7, 2023, pp. 271–284. https://doi.org/10.1007/s42064-022-0157-x

    2023 doi

  13. [22]

    State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit,

    Gim, D.-W., and Alfriend, K. T., “State Transition Matrix of Relative Motion for the Perturbed Noncircular Reference Orbit,” Journal of Guidance, Control, and Dynamics, Vol. 26, No. 6, 2003, pp. 956–971. https://doi.org/10.2514/2.6924

    2003 doi

  14. [23]

    Nonlinear mapping of Gaussian statistics: Theory and applications to spacecraft trajectory design,

    Park, R. S., and Scheeres, D. J., “Nonlinear mapping of Gaussian statistics: Theory and applications to spacecraft trajectory design,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 6, 2006, pp. 1367–1375. https://doi.org/10.2514/1.20177

    2006 doi

  15. [24]

    Orbit propagation in irregular and uncertain gravity field using differential algebra,

    Feng, J., Armellin, R., and Hou, X., “Orbit propagation in irregular and uncertain gravity field using differential algebra,”Acta Astronautica, Vol. 161, 2019, pp. 338–347. https://doi.org/10.1016/j.actaastro.2019.05.045

    2019 doi

  16. [25]

    A new method for the nonlinear transformation of means and covariances in filters and estimators,

    Julier, S., Uhlmann, J., and Durrant-Whyte, H. F., “A new method for the nonlinear transformation of means and covariances in filters and estimators,”IEEE Transactions on Automatic Control, Vol. 45, No. 3, 2000, pp. 477–482. https://doi.org/10.1109/9. 847726

    doi

  17. [26]

    Performance evaluation of UKF-based nonlinear filtering,

    Xiong, K., Zhang, H., and Chan, C., “Performance evaluation of UKF-based nonlinear filtering,”Automatica, Vol. 42, No. 2, 2006, pp. 261–270. https://doi.org/10.1016/j.automatica.2005.10.004

    2006 doi

  18. [27]

    Conjugate unscented transformation-based approach for accurate conjunction analysis,

    Adurthi, N., and Singla, P., “Conjugate unscented transformation-based approach for accurate conjunction analysis,”Journal of Guidance, Control, and Dynamics, Vol. 38, No. 9, 2015, pp. 1642–1658. https://doi.org/10.2514/1.G001027

    2015 doi

  19. [28]

    Conjugate Unscented Transformation: Applications to Estimation and Control,

    Adurthi, N., Singla, P., and Singh, T., “Conjugate Unscented Transformation: Applications to Estimation and Control,”Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME, Vol. 140, No. 3, 2018. https://doi.org/10.1115/1. 4037783

    2018 doi

  20. [29]

    Conjugateunscentedtransformation–baseduncertaintyanalysisofenergyharvesters,

    Nanda,A.,Singla,P.,andKarami,M.A.,“Conjugateunscentedtransformation–baseduncertaintyanalysisofenergyharvesters,” Journal of Intelligent Material Systems and Structures , Vol. 29, No. 18, 2018, pp. 3614–3633. https://doi.org/10.1177/ 1045389X18798945

    2018

  21. [30]

    Asteroid close encounters characterization using differential algebra: the case of Apophis,

    Armellin, R., Di Lizia, P., Bernelli-Zazzera, F., and Berz, M., “Asteroid close encounters characterization using differential algebra: the case of Apophis,” Celestial Mechanics and Dynamical Astronomy , Vol. 107, No. 4, 2010, pp. 451–470. https://doi.org/10.1007/s10569-010-9283-5

    2010 doi

  22. [31]

    Nonlinear semi-analytic methods for trajectory estimation,

    Park, R. S., and Scheeres, D. J., “Nonlinear semi-analytic methods for trajectory estimation,”Journal of Guidance, Control, and Dynamics, Vol. 30, No. 6, 2007, pp. 1668–1676. https://doi.org/10.2514/1.29106. 35

    2007 doi

  23. [32]

    A high order method for estimation of dynamic systems,

    Majji, M., Junkins, J. L., and Turner, J. D., “A high order method for estimation of dynamic systems,”The Journal of the Astronautical Sciences, Vol. 56, No. 3, 2008, pp. 401–440. https://doi.org/10.1007/BF03256560

    2008 doi

  24. [33]

    Modern Map Methods in Particle Beam Physics,

    Berz, M., and Hawkes, P. W., “Modern Map Methods in Particle Beam Physics,”Advances in Imaging and Electron Physics, Vol. 108, 1999, pp. 81–117. https://doi.org/10.1016/S1076-5670(08)70228-3

    1999 doi

  25. [35]

    ToolstodetectstructuresindynamicalsystemsusingJetTransport,

    Pérez-Palau,D.,Masdemont,J.J.,andGómez,G.,“ToolstodetectstructuresindynamicalsystemsusingJetTransport,” Celestial Mechanics and Dynamical Astronomy, Vol. 123, No. 3, 2015, pp. 239–262. https://doi.org/10.1007/s10569-015-9634-3

    2015 doi

  26. [36]

    Long-Term Fuel-Optimal Collision Avoidance Maneuvers with Station-Keeping Constraints,

    Pavanello, Z., Pirovano, L., and Armellin, R., “Long-Term Fuel-Optimal Collision Avoidance Maneuvers with Station-Keeping Constraints,” Journal of Guidance, Control, and Dynamics, Vol. 47, No. 9, 2024, pp. 1855–1871. https://doi.org/10.2514/1. G007839

    2024 doi

  27. [37]

    Nonlinear semi-analytical uncertainty propagation of trajectory under impulsive maneuvers,

    Yang, Z., Luo, Y., and Zhang, J., “Nonlinear semi-analytical uncertainty propagation of trajectory under impulsive maneuvers,” Astrodynamics, Vol. 3, No. 1, 2019, pp. 61–77. https://doi.org/10.1007/s42064-018-0036-7

    2019 doi

  28. [38]

    Reduced Nonlinear Model for Orbit Uncertainty Propagation and Estimation,

    Roa, J., and Park, R. S., “Reduced Nonlinear Model for Orbit Uncertainty Propagation and Estimation,”Journal of Guidance, Control, and Dynamics, Vol. 44, No. 9, 2021, pp. 1578–1592. https://doi.org/10.2514/1.G005519

    2021 doi

  29. [39]

    Neural Network-Based Method for Orbit Uncertainty Propagation and Estimation,

    Zhou, X., Qiao, D., and Li, X., “Neural Network-Based Method for Orbit Uncertainty Propagation and Estimation,”IEEE Transactions on Aerospace and Electronic Systems, Vol. 60, No. 1, 2024, pp. 1176–1193. https://doi.org/10.1109/TAES.2023. 3332566

    2024 doi

  30. [40]

    Directional State Transition Tensors for Capturing Dominant Nonlinear Effects in Orbital Dynamics,

    Boone, S., and McMahon, J., “Directional State Transition Tensors for Capturing Dominant Nonlinear Effects in Orbital Dynamics,” Journal of Guidance, Control, and Dynamics,Vol.46,No.3,2022,pp.431–442. https://doi.org/10.2514/1.G006910

    2022 doi

  31. [41]

    Efficient Nonlinear Spacecraft Navigation Using Directional State Transition Tensors,

    Boodram, O., Boone, S., and McMahon, J., “Efficient Nonlinear Spacecraft Navigation Using Directional State Transition Tensors,” AAS Space Flight Mechanics, 2022, pp. AAS 22–670

    2022

  32. [42]

    Efficient Nonlinear Filtering Methods for Near Rectilinear Halo Orbit Navigation,

    Boone, S., Boodram, O., and McMahon, J., “Efficient Nonlinear Filtering Methods for Near Rectilinear Halo Orbit Navigation,” AAS Guidance, Navigation and Control Conference, Breckenridge, CO, 2022, pp. AAS 23–065

    2022

  33. [43]

    An Efficient Approximation of the Second-Order Extended Kalman Filter for a Class of Nonlinear Systems,

    Boone, S., and McMahon, J., “An Efficient Approximation of the Second-Order Extended Kalman Filter for a Class of Nonlinear Systems,”2024 European Control Conference (ECC), 2024, pp. 3533–3538. https://doi.org/10.23919/ECC64448. 2024.10591212

    2024

  34. [44]

    Configuration uncertainty propagation of gravitational-wave observatory using a directional state transition tensor,

    Qiao, D., Zhou, X., and Li, X., “Configuration uncertainty propagation of gravitational-wave observatory using a directional state transition tensor,”Chinese Journal of Aeronautics, Vol. 1, 2024, pp. 1–20. https://doi.org/10.1016/j.cja.2024.06.015. 36

    2024 doi

  35. [45]

    Simplified calculation of eigenvector derivatives,

    Nelson, R. B., “Simplified calculation of eigenvector derivatives,”AIAA Journal, Vol. 14, No. 9, 1976, pp. 1201–1205. https://doi.org/10.2514/3.7211

    1976 doi

  36. [46]

    Eigenvector derivatives with repeated eigenvalues,

    Dailey, R. L., “Eigenvector derivatives with repeated eigenvalues,”AIAA Journal, Vol. 27, No. 4, 1989, pp. 486–491. https://doi.org/10.2514/3.10137

    1989 doi

  37. [47]

    Gradients for Large Structural Models with Repeated Frequencies,

    Ojalvo, I. U., “Gradients for Large Structural Models with Repeated Frequencies,”SAE Aerospace Technology Conference and Exposition, SAE International, 1986. https://doi.org//10.4271/861789

    1986 doi

  38. [48]

    Investigating temporary capture in the Sun–Jupiter three-body system via Lagrangian coherent structures,

    Li, Z., Qiao, D., and Li, X., “Investigating temporary capture in the Sun–Jupiter three-body system via Lagrangian coherent structures,” Monthly Notices of the Royal Astronomical Society , Vol. 534, No. 1, 2024, pp. 902–917. https: //doi.org/10.1093/mnras/stae2122

    2024 doi

  39. [49]

    Adaptive Order-Switching Kalman Filter for Orbit Determination Using Deep-Neural- Network-Based Nonlinearity Detection,

    Zhou, X., Qiao, D., and Li, X., “Adaptive Order-Switching Kalman Filter for Orbit Determination Using Deep-Neural- Network-Based Nonlinearity Detection,” Journal of Spacecraft and Rockets , Vol. 60, No. 6, 2023, pp. 1724–1741. https://doi.org/10.2514/1.A35639. 37

    2023 doi

Pith tools

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