Pith. sign in

REVIEW 4 major objections 5 minor 38 references

A polynomial-based Monte Carlo approach for estimating long-term collision probabilities

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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

Pith's one-line read This paper claims that long-term satellite collision probability — the slow, bending encounters of GEO neighbours and rendezvous — becomes computable by sampling the initial uncertainty and evaluating Taylor polynomials of the miss distance

desk verdict DA/ADS polynomial MC for long-term collision probability: real speed-ups and a broad empirical match to MC, but the 0.5% error claim is overstated and the root-finding section is not rigorous. read the letter →

arxiv 2509.07607 v1 pith:HMIKB4TC submitted 2025-09-09 physics.space-ph

Andrea Zollo , Cristina Parigini , Roberto Armellin , Juan Félix San Juan Díaz , Annarita Trombetta , Ralph Kahle This is my paper
classification physics.space-ph PACS 95.10.Ce
keywords collisionprobabilitylong-termencounterdifferentialalgebraautomaticdomainsplittingTaylorpolynomialMonteCarlosimulationsatelliteconjunctionrootfinding
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

The paper aims to make long-term satellite collision risk cheap, targeting the slow encounters between neighbouring GEO satellites, rendezvous, and formation flights, where the relative trajectory bends non-linearly and classical Monte Carlo propagation becomes expensive. Its claim is that a single Differential Algebra integration of the relative dynamics, subdivided into accuracy-controlled Taylor patches, captures the whole initial uncertainty set at once; each Monte Carlo sample then costs only a polynomial evaluation, and a collision is detected by finding real roots of a one-dimensional time polynomial for the squared miss distance minus the hard-body-radius squared. Across ten literature benchmark cases and one real GEO conjunction, the method reproduces Monte Carlo probabilities with errors never exceeding 0.5% and computational gains from roughly 150% to nearly 60,000%. If the claim holds, long-term conjunction screening becomes fast enough for operational use across many satellites, and the same reduction — sample, evaluate a polynomial, count roots — could serve any threshold-crossing probability problem.

What carries the argument

The load-bearing object is the patched 7D Taylor manifold: the relative state vector expressed as a high-order Taylor polynomial in scaled time τ (normalized to [−1,1]) and six normalized initial-condition deviations δx₀, aligned to the eigen-axes of the initial combined covariance. Each patch is generated by a single Differential Algebra (DA) integration of the relative dynamics, and the Automatic Domain Splitting (ADS) algorithm halves any patch whose Taylor expansion departs from the true flow beyond a user tolerance ε, producing a piecewise polynomial representation of the whole uncertainty set. The second key element is the reduction of each sampled realization to a one-dimensional poly

What would settle it

A targeted stress test would put many samples' minimum miss distance within the Taylor error band of the hard-body radius — a strong non-linear encounter where true miss distances cluster near HBR — then compare DA-derived hit/miss classifications against a very large reference Monte Carlo (10⁸ propagations) on identical initial conditions and dynamics. If the polynomial sign near zero disagrees with true propagation for any measurable fraction of samples, or the estimated Pc jumps discontinuously as the ADS tolerance ε is tightened, the polynomial-fidelity premise fails.

Watch

Extended reading notes

Core claim

The paper claims that collision probability — a six-dimensional integral over initial relative states — reduces to sampling plus polynomial evaluation. A patched 7D Taylor manifold represents the relative flow as high-order Taylor polynomials in time and six initial-condition deviations, one Differential Algebra integration per patch, with Automatic Domain Splitting enforcing accuracy. Each sampled realization collapses to a univariate time polynomial in d² − HBR²; a real root means a hit, found via Budan-Fourier sign counts or the Sturm algorithm. The flagship case gives Pc = 2.1783e-01, identical to the Monte Carlo reference, and the real GEO case matches MC with 0.0% error across three ac

Load-bearing premise

The load-bearing premise is that the Taylor patches representing the dynamics keep the expression d² − HBR² faithful everywhere inside the 5-sigma domain, so every polynomial root corresponds to a real dip below the hard-body radius and no real dip is missed; the paper validates this empirically on its test cases but supplies no formal bound linking the accuracy tolerance to correct hit/miss classification.

Editorial extensions

If this is right

  • Long-term conjunctions — GEO neighbours, rendezvous, formation flight — can be screened with Monte Carlo-level accuracy at a fraction of the runtime, with the advantage growing as the dynamical model becomes more complex.
  • Multiple close approaches inside one screening window are captured automatically: test case #1 shows Pc accumulating at two distinct epochs, and the real GEO case shows three accumulation windows from a single polynomial sweep.
  • The method needs no Gaussian assumption after t₀ and no constant-covariance or straight-line assumptions, so it covers regimes the classic short-term formulas cannot.
  • Sample evaluation parallelizes cleanly across CPU cores, and the DA integration cost is negligible compared with sample processing, making operational, multi-satellite screening plausible.
  • For essentially instantaneous short-term encounters the method still matches the reference (error ≤ 0.5%) but needs many domain splits near TCA; the paper notes dedicated 2D methods remain more suitable there.

Reading between the lines

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

  • The reduction is not specific to satellite conjunctions: any first-passage probability where the event is 'a noisy trajectory dips below a threshold within a time window' — re-entry debris casualty risk, structural fatigue, relay trips — could reuse the sample-then-evaluate-polynomials flow, since the polynomial root gives the crossing time, not just that a crossing occurred.
  • The 5-sigma truncation of the initial domain is the one systematic error the authors acknowledge; an importance-sampling or tail-patching extension that covers samples beyond 5σ would give operational users full-support coverage without slowing the bulk computation.
  • Because the speed advantage widens with dynamical complexity, coupling the method with high-fidelity perturbation models — solar radiation pressure, third-body attraction, drag — is where the largest operational gains should appear, since that is exactly where brute-force Monte Carlo is most expensive.
  • A cheap safety layer suggests itself: re-propagate with true dynamics only those samples whose polynomial minimum lies within the ADS tolerance band of the hard-body radius, giving the hit/miss classification a formal safety margin without full Monte Carlo cost.
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 / 5 minor

Summary. The paper proposes a Monte Carlo method for long-term conjunction probability in which the relative-state ODE flow is represented as a patchwork of high-order Taylor polynomials computed with Differential Algebra and Automatic Domain Splitting (ADS). After sampling the initial relative-state PDF, each sample is evaluated in the polynomial surrogates, reducing hit detection to root finding for 1-D time polynomials f_d = d^2 - HBR^2. Pc is the fraction of samples with a real root in the screening interval. The method is validated against an in-house MC tool on ten benchmark cases spanning Clohessy-Wiltshire, Yamanaka-Ankersen, and D'Amico ROE dynamics, plus a real GEO conjunction; reported Pc errors are mostly below 0.02% except short-term case #3 (~0.4-0.5%), with speedups from ~150% to nearly 60,000%.

Significance. If correct, this is a useful extension of DA-based Pc computation to long-term, non-linear encounters, retaining MC flexibility (non-Gaussian initial PDFs, velocity uncertainty, multiple conjunctions) while drastically reducing computational cost. The validation is broad, the real GEO example is valuable, and the comparison against an independent in-house MC tool is a clear strength. The paper does not ship code, but the methodology is reproducible in principle given the stated workflow. However, the lack of a formal link between ADS truncation error and the binary collision indicator, together with the flawed description of the Budan-Fourier root test, means the theoretical basis needs repair; the empirical claims are plausible but not yet fully supported.

major comments (4)
  1. [Section 3.3, Eq. (13)] The text equates Budan-Fourier with counting sign changes of the polynomial coefficients and uses 'zero sign changes' to conclude no root in the interval. This is not correct for a polynomial expanded about an arbitrary interior point. For example, f(τ)=(τ+0.1)(τ+0.2), expanded about τ=0, has coefficient signs (+,+,+), hence zero sign changes, yet has two roots in (-1,1). The theorem requires comparing sign variations of the derivative sequence at the two interval endpoints (or transforming the interval to [0,1]). If the implementation follows the text, collisions can be missed; if it does not, the description must be corrected and the actual test (e.g., Sturm on every interval) stated.
  2. [Sections 3.2/3.3] ADS controls the truncation error of the Taylor map for the state to a user tolerance ε, but no bound is derived for the induced error in f_d = d^2 - HBR^2 or, more importantly, for the sign/root structure of f_d on the patch. Because the collision indicator is discontinuous (a root exists or not), a state error of order ε can flip the indicator when the true miss distance is within ε of HBR. The unbiasedness of the Pc estimator is therefore not established; the validation set consists overwhelmingly of high-Pc cases. Please provide a formal error bound or a targeted numerical study with near-tangent, low-Pc conjunctions.
  3. [Section 4.2, Table 2] The text states 'the error with respect to the reference never exceeds 0.0176%' and later 'In case #3 ... the error ... never exceeds 0.5%'. Table 2 reports D'Amico ROEs case #3 error as 0.5070%, and the earlier 0.0176% bound excludes case #3. This is internally inconsistent. Please correct the summary claims and either explain the 0.5070% case or adjust the stated bound.
  4. [Section 4.2 and Eq. (9)] The method depends on several free parameters that are never specified: Taylor expansion order, ADS splitting tolerance ε, and the 5σ truncation threshold Δx0. Table 2 reports only ADS integration time, not the parameter values. The 5σ truncation is acknowledged to discard samples, which can bias Pc in the low-probability tail regime that is operationally relevant. Report the parameter values, add a sensitivity analysis, and discuss how the truncation error is controlled for small Pc.
minor comments (5)
  1. [Section 3.1] There are duplicate equation numbers: two equations are labelled (9). Renumber the equations.
  2. [Section 3.1] Incomplete sentence immediately before Eq. (10): 'That is a vector of high order polynomials...' needs to be completed.
  3. [References] References [27] and [31] are both Whitney, 'Differentiable Manifolds.' Remove the duplicate.
  4. [Sections 3.1/3.2] The claim that 'only a single integration is required' is misleading because ADS re-expansion around new centers requires further DA integrations; the non-negligible 'ADS integration time' columns in Table 2 confirm this. Clarify the wording.
  5. [Table 2 and Fig. 9] Minor typographical issues: 'Pc Cmp. Time' should be 'Pc Comp. Time'; in Fig. 9, the percentage gain axis label and the log-scale time axis could be more clearly separated.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Pc computed from sampled initial PDF and polynomial surrogates; MC used only as external benchmark.

full rationale

The derivation chain is self-contained: the paper defines the collision event through the initial set V (Eq. 4), samples the initial Gaussian PDF (Eq. 5), propagates a DA/ADS polynomial manifold (Eqs. 10-13), and counts samples whose one-dimensional miss-distance polynomial has a real root (Section 3.3). No parameter is fitted to the reference Monte Carlo results; the MC simulations serve only as an external benchmark, using standard literature test cases [19] plus a real conjunction. The self-citations [24]-[26] provide the underlying DA and ADS computational machinery, but they are not used to certify the numerical agreement or to preclude alternative methods; they are normal methodological references. The paper's accuracy claims rest on the explicit comparison in Table 2, which is an independent check rather than a circular target. The identified weaknesses—lack of a formal error bound transferring the ADS tolerance to the sign/root structure of f_d, and the 5-sigma truncation of outlying samples—are accuracy and verification concerns, not circular reductions. The table even shows a minor inconsistency with the text's 'never exceeds 0.5%' statement (D'Amico ROEs case #3 is 0.507%), but that is a correctness issue, not a circularity. Therefore no circular step is exhibited.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities or conserved quantities. It relies on standard mathematical theorems and domain assumptions about state independence, Gaussian initial uncertainty, and the choice of relative dynamics models. The free parameters are numerical accuracy and truncation settings, not fitted quantities.

free parameters (3)
  • ADS splitting tolerance (epsilon) = not reported
    User-defined accuracy threshold for the Taylor expansions; it controls how finely the initial domain is split and therefore the accuracy and runtime of the method. It is not fitted to the reference MC results.
  • Initial deviation truncation threshold = 5 standard deviations
    Samples whose initial relative state falls outside 5-sigma of the mean are discarded, which truncates the PDF support and can bias the computed Pc. This is a practical modeling choice.
  • Taylor expansion order = not reported
    The polynomial order used in the DA expansions is not stated numerically, but it directly affects the accuracy of the manifold and the computational cost.
assumptions (5)
  • domain assumption Primary and secondary object states are statistically independent with zero cross-covariance.
    Introduced in Section 2, Eq. (2). This is the standard assumption in conjunction assessment and is stated explicitly.
  • domain assumption The initial relative state is Gaussian with known mean and covariance.
    Section 2, Eq. (5). The paper notes the method can handle arbitrary distributions but uses Gaussianity for the presented tests.
  • domain assumption The chosen analytical relative dynamics models (Clohessy-Wiltshire, Yamanaka-Ankersen, D'Amico ROEs with J2) faithfully describe the motion during the encounter.
    Section 4 uses these models both to back-map TCA conditions to t0 and to propagate the DA manifold. The validation therefore tests the polynomial approximation, not the fidelity of the dynamics models.
  • domain assumption The ADS algorithm ensures the Taylor map is accurate to within a user-specified tolerance on the whole domain.
    Section 3.2 relies on the automatic domain splitting technique from Wittig et al. [26]. The paper does not derive an a priori error bound for the distance function.
  • standard math Budan-Fourier and Sturm theorems correctly isolate real roots of univariate polynomials.
    Section 3.3 invokes these classical theorems for root detection in the time polynomials. The paper's description of the Budan-Fourier theorem is imprecise, but the invoked mathematical results are standard.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A polynomial-based Monte Carlo approach for estimating long-term collision probabilities." pith.science (2026). https://pith.science/paper/HMIKB4TC

@misc{pith2026250907607,
  author       = {Pith},
  title        = {Pith review of: A polynomial-based Monte Carlo approach for estimating long-term collision probabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HMIKB4TC}},
  note         = {Machine review of arXiv:2509.07607}
}
read the original abstract

This paper introduces a versatile approach for computing the risk of collision specifically tailored for scenarios featuring low relative encounter velocities, but with potential applicability across a wide range of situations. The technique employs Differential Algebra (DA) to express the non-linear dynamical flow of the initial distribution in the primary-secondary objects relative motion through high-order Taylor polynomials. The entire initial uncertainty set is subdivided into subsets through Automatic Domain Splitting (ADS) techniques to control the accuracy of the Taylor expansions. The methodology samples the initial conditions of the relative state and evaluates the polynomial expansions for each sample while retaining their temporal dependency. The classical numerical integration of the initial statistics over the set of conditions for which a collision occurs is thus reduced to an evaluation of mono-dimensional time polynomials. Specifically, samples reaching a relative distance below a critical value are identified along with the time at which this occurs. The approach is tested against a Monte Carlo (MC) simulation for various literature test cases, yielding accurate results and a consistent gain in computational time.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

38 extracted references · 35 canonical work pages

  1. [1]

    ESA’s annual space environment report

    ESOC (European Space Operations Centre) , “ ESA’s annual space environment report ”, ESA Space Debris Office (2025), ESA TR GEN-DBLOG-00288-OPS-SD

    work page 2025

  2. [2]

    Space Debris – Models and Risk Analysis

    Klinkrad, H., “Space Debris – Models and Risk Analysis”. Springer, 2006, Berlin Heidelberg

    work page 2006

  3. [3]

    Short-term vs. long -term spacecraft encounters

    Chan, K. , “Short-term vs. long -term spacecraft encounters. ”, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 2004, Providence, RI, Paper AIAA-2004-5460 22

    work page 2004

  4. [4]

    Parametric analysis of orbital debris collision probability and maneuver rate for space vehicles

    Foster, J. A., Estes, H. S. “Parametric analysis of orbital debris collision probability and maneuver rate for space vehicles.” Technical Report. NASA JSC, 1992

    work page 1992

  5. [5]

    General method for calculating satellite collision probability

    Patera, R. P. “General method for calculating satellite collision probability.” Journal of Guidance, Control, and Dynamics, 2001, 24(4): 716–722

    work page 2001

  6. [6]

    Calculating collision probability for arbitrary space vehicle shapes via numerical quadrature

    Patera, R. P. “Calculating collision probability for arbitrary space vehicle shapes via numerical quadrature.” Journal of Guidance, Control, and Dynamics, 2005, 28(6): 1326–1328

    work page 2005

  7. [7]

    Probability of collision error analysis

    Alfriend, K. T., Akella, M. R., Frisbee, J., Foster, J. L., Deok -Jin, L., Wilkins, M. “Probability of collision error analysis.” Space Debris, 1999, 1(1): 21–35

    work page 1999

  8. [8]

    A numerical implementation of spherical object collision probability

    Alfano, S. “A numerical implementation of spherical object collision probability.” The Journal of the Astronautical Sciences, 2005, 53(1): 103–109

    work page 2005

Show all 38 references
  1. [9]

    Chan, F. K. Spacecraft Collision Probability. Reston, VA, USA: Aerospace Press, 2008

    2008

  2. [10]

    Chan, K.,.”Collision probability analyses for earth orbiting satellites ”. Adv. Astronaut. Sci. 96, 1033 – 1048, 1997

    1997

  3. [11]

    Fast and accurate computation of orbital collision probability for short-term encounters

    Serra, R., Arzelier, D., Joldes, M., Lasserre, J. B., Rondepierre, A., Salvy, B. “Fast and accurate computation of orbital collision probability for short-term encounters.” Journal of Guidance, Control, and Dynamics, 2016, 39(5): 1009–1021

    2016

  4. [12]

    Satellite collision probability for nonlinear relative motion

    Patera, R. P. “Satellite collision probability for nonlinear relative motion.” Journal of Guidance, Control, and Dynamics, 2003, 26(5): 728–733

    2003

  5. [13]

    Collision probability for larger bodies having nonlinear relative motion

    Patera, R. P. “Collision probability for larger bodies having nonlinear relative motion.” Journal of Guidance, Control, and Dynamics, 2006, 29(6): 1468–1472

    2006

  6. [14]

    Addressing nonlinear relative motion for spacecraft collision probability

    Alfano, S. “Addressing nonlinear relative motion for spacecraft collision probability.” In: Proceedings of the AIAA/AAS Astrodynamics Specialist Conference and Exhibit, 2006: 6760

    2006

  7. [15]

    Hovering collision probability

    F.K. Chan. “Hovering collision probability.” In AAS/AIAA Space Flight Mechanics Meeting, number AAS 15-234, Williamsburg, VA, USA, January 2015

    2015

  8. [16]

    Spacecraft collision probability for long -term encounters

    K. Chan. “Spacecraft collision probability for long -term encounters.” Number AAS 03 -549, Big Sky, Montana, USA, 2003

    2003

  9. [17]

    Including Velocity Uncertainty in the Probability of Collision between Space Objects

    V.T. Coppola. “Including Velocity Uncertainty in the Probability of Collision between Space Objects.” Advances in the Astronautical Sciences, 143, 2012

    2012

  10. [18]

    „Poly nomial superlevel set approximation of swept -volume for computing collision probability in space encounters.” 2021

    Denis Arzelier, Florent Bréhard, Mioara Joldeş, Jean -Bernard Lasserre, Sohie Laurens, et al. „Poly nomial superlevel set approximation of swept -volume for computing collision probability in space encounters.” 2021. hal-03158347

    2021

  11. [19]

    Advances in the Astronautical Sciences, 134:2007–2024, 2009

    Alfano S., Satellite conjunction Monte Carlo analysis. Advances in the Astronautical Sciences, 134:2007–2024, 2009

    2007

  12. [20]

    High Fidelity Collision Probabilities Estimated Using Brute Force Monte Carlo Simulations

    Hall, D.T., Casali, S.J., Johnson, L.C., Skrehart, B.B., & Baars, L.G. , “High Fidelity Collision Probabilities Estimated Using Brute Force Monte Carlo Simulations ”, 2018, https://api.semanticscholar.org/CorpusID:126009860

    2018

  13. [21]

    Estimation of small failure probabilities in high dimensions by subset simulation

    Au, S.-K., Beck, J.L.,. “Estimation of small failure probabilities in high dimensions by subset simulation.” Probab. Eng. Mech. 16 (4), 263–277, 2001

    2001

  14. [22]

    Reliability of structures in high dimensions, part I: algorithms and applications

    Koutsourelakis P.S., Pradlwarter H.J., Schuëller G.I.,“Reliability of structures in high dimensions, part I: algorithms and applications”, Probabilistic Engineering Mechanics, Volume 19, Issue 4, 2004, Pages 23 409-417, ISSN 0266-8920, https://doi.org/10.1016/j.probengmech.2004.05.001

    2004 doi

  15. [23]

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

    Armellin, R., Di Lizia, P., Bernelli Zazzera, F., Berz, M. “Asteroid close encounters characterization using differential algebra: the case of Apophis.” Celest. Mech. Dyn. Astron. 107, 451–470 (2010)

    2010

  16. [24]

    Morselli A., Armellin R., Di Lizia P., Bernelli Zazzera F., ” A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation ”, Advances in Space Research, Volume 55, Issue 1, 2015, Pages 311-333, ISSN 0273-1177, https://doi.org/10.1016/j....

    2015 doi

  17. [25]

    Long -term collision probability computation through high order polynomials evaluation

    A. Zollo, C. Parigini, R. Armellin, J. F. San Juan Díaz, S. Aida, R. Kahle, “Long -term collision probability computation through high order polynomials evaluation”, 29th International Symposium on Space Flight Dynamics (ISSFD), Darmstadt, Germany, 22-26th April 2024

    2024

  18. [26]

    Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting,

    A. Wittig, P. Di Lizia, R. Armellin, K. Makino, F. Bernelli -Zazzera, and M. Berz, “Propagation of large uncertainty sets in orbital dynamics by automatic domain splitting,” Celestial Mechanics and Dynamical Astronomy, Vol. 122, No. 3, 2015, pp. 239–261

    2015

  19. [28]

    Conjunction risk assessment and avoidance maneuver planning tools

    Aida S., “Conjunction risk assessment and avoidance maneuver planning tools. ”, 6th International Conference on Astrodynamics Tools and Techniques, Darmstadt, Germany, March 2016

    2016

  20. [29]

    H.; Wiltshire, R

    Clohessy, W. H.; Wiltshire, R. S. (1960). Terminal Guidance System for Satellite Rendezvous. Journal of the Aerospace Sciences. 27 (9): 653–658. doi:10.2514/8.8704

    1960 doi

  21. [30]

    Determining Satellite Close Approaches,

    Alfano, S. "Determining Satellite Close Approaches," Journal of the Astronautical Sciences, Vol. 41, No. 2, April-June 1993, pp. 217-225

    1993

  22. [31]

    Differentiable Manifolds

    Whitney, H., “Differentiable Manifolds.” Annals of Mathematics, vol. 37, no. 3, 1936, pp. 645 –80., https://doi.org/10.2307/1968482

    1936 doi

  23. [32]

    Caleb, T., Losacco, M., Fossà, A. et al. ,” Differential algebra methods applied to continuous abacus generation and bifurcation detection: application to periodic families of the Earth –Moon system”. Nonlinear Dynamics, Volume 111, pp. 9721–9740, 2023, https://doi.org/10.1007...

    2023 doi

  24. [33]

    Sur l'usage du théorème de Descartes dans la recherche des limites des racines

    Fourier, Jean Baptiste Joseph (1820). "Sur l'usage du théorème de Descartes dans la recherche des limites des racines". Bulletin des Sciences, par la Société Philomatique de Paris: 156–165

  25. [34]

    Mémoire sur la résolution des équations numériques

    Sturm, Jacques Charles François (1829). "Mémoire sur la résolution des équations numériques". Bulletin des Sciences de Férussac. 11: 419–425

  26. [35]

    Fitting atmospheric parameters using parabolic blending

    Alfano, S. and Gandhi, A. M., “Fitting atmospheric parameters using parabolic blending”, Journal of Aircraft, 1990, https://doi.org/10.2514/3.45986Alfano, S

    1990 doi

  27. [36]

    and Ankersen, F .,”New State Transition Matrix for Relative Motion on an Arbitrary Elliptical Orbit ”, Journal of Guidance, Control, and Dynamics , V olume 25 , pp

    Yamanaka, K . and Ankersen, F .,”New State Transition Matrix for Relative Motion on an Arbitrary Elliptical Orbit ”, Journal of Guidance, Control, and Dynamics , V olume 25 , pp. 60-66, 2002, https://doi.org/10.2514/2.4875

    2002 doi

  28. [37]

    D'Amico, S.,” Autonomous formation flying in low earth orbit.” Ridderkerk: Ridderprint bv, 2010. 209 p

    2010

  29. [38]

    Comprehensive Survey and Assessment of Spacecraft Relative Motion Dynamics Models

    Sullivan, J., Grimberg, S. and D’Amico, S., “Comprehensive Survey and Assessment of Spacecraft Relative Motion Dynamics Models ”, Journal of Guidance, Control, and Dynamics , Volume 40, 2017, https://doi.org/10.2514/1.G002309 24

    2017 doi

  30. [39]

    Closing the gap between SGP4 and high-precision propagation via differentiable programming

    Acciarini, G., Atılım, G. B., Izzo, D., “Closing the gap between SGP4 and high-precision propagation via differentiable programming”, Acta Astronautica, Volume 226, Part 1, 2025, Pages 694-701, ISSN 0094- 5765, https://doi.org/10.1016/j.actaastro.2024.10.063

    2025 doi

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.