Scientific Computing Group (GRUCACI) University of La ... · The continuous development and upgrade of orbit ... Application of SAPO, the GMV’s Semi-Analytical Orbit Propagator,

4TH INTERNATIONAL WORKSHOP ON KEY TOPICS IN ORBIT PROPAGATION

APPLIED TO SPACE SITUATIONAL AWARENESS

Scientific Computing Group (GRUCACI) University of La Rioja, Logroño, Spain

April 24‐26, 2019

PROGRAM & ABSTRACT BOOK

Organization:

Sponsorship:

KePASSA 2019. 24‐26 Apr. 2019 in Logroño, Spain 2

PREFACE

For its fourth edition, KePASSA goes back to Logroño, where the series of

workshops was kicked off in 2014.

The workshop aims to gather worldwide experts in the field of orbit propagation

who deliver talks of top-notch scientific content in an informal, friendly and relaxed

environment.

Orbit propagation, the main topic of the workshop, is one of the most challenging

and complex aspects of astronautics and will play a key role in ensuring the

sustainable use of space. The continuous development and upgrade of orbit

propagation techniques is instrumental for tackling most of the SSA challenges,

e.g. those posed by the imminent realization of mega-constellations, the creation

of sensors capable of tracking hundreds of thousands of objects, and the

implementation of a space traffic management and control system.

To allow researchers to present last-minute results KePASSA is a presentation-

only workshop, although proceedings are edited when a sufficient number of

authors requires it. Stimulating scientific discussions, creation of partnership and

collaborations will be facilitated by the warm hospitality of La Rioja, its people, its

food and, importantly, its wine. We are looking forward to meeting you soon and

celebrating together another successful KePASSA meeting!


AIMS AND SCOPE

The workshop will be an opportunity to showcase the progress made since our last meeting in ESTEC in the following topics:

Analytical, semi-analytical, and numerical propagation methods

Hybrid and statistical methods

Uncertainty quantification and propagation

State vector representation, orbital elements/coordinates

Long-term and short-term propagation

Resonances and chaos

Series expansions

Special functions

Non-gravitational perturbations

High fidelity models

Symbolic computation

Software packages for orbit propagation

Third body and time dependence in the analytical method

End-of-life disposal

Collision probability computation

Planetary protection

High-Earth Orbits and Highly-Elliptical Orbits

Relative dynamics

SPECIAL SESSIONS

Space debris modelling and removal

Orbit determination and cataloguing

Propagation techniques and applications

Orbital conjuctions

Scientific and educational software tools

Analytical and semi-analytical methods

Dynamics modelling and analysis


ORGANIZING COMMITTEE

Scientific Committee

Armellin, Roberto (University of Surrey, UK)

Cefola, Paul (State University of New York at Buffalo, USA)

Deleflie, Florent (IMCCE/GRGS, Observatoire de Paris, France)

Hautesserres, Denis (CNES, France)

Lara, Martín (GRUCACI-UR, Spain)

Peláez, Jesús (SDG-UPM, Spain)

San-Juan, Juan Félix (GRUCACI-UR, Spain)

Wnuk, Edwin (AMU Observatory, Poland)

Local Committee

Armellin, Roberto (University of Surrey, UK)

Carrillo, Hans (GRUCACI-UR, Spain)

Hautesserres, Denis (CNES, France)

Lara, Martín (GRUCACI-UR, Spain)

López, Rosario (GRUCACI-UR, Spain)

Pérez, Iván (GRUCACI-UR, Spain)

San-Juan, Juan Félix (GRUCACI-UR, Spain)

San-Martín, Montse (GRUCACI-UG, Spain)

Segura, Edna (GRUCACI-UR, Spain)

Vergara, Eliseo (GRUCACI-UR, Spain)


KePASSA 2019 (April 24-26, 2019)

Programme

Wednesday 24

9:00 – 9:50 Registration

9:15 – 10:00 Opening

10:15 – 11:00 Plenary: Hodei Urrutxua – Regularization and Error Propagation in Orbit Predictions

11:00 – 11:20 Coffee break

11:20 – 13:00 Space Debris Modelling and Removal Chair: E.M. Alessi

11:20 – 11:40 C. Bombardelli, G. Falco, D. Amato and A. Rosengren – Measuring space occupancy of LEO satellites

11:40 – 12:00 A. Petit and D. Casanova – Comparison of several methods used to create a synthetic population of the space debris in the GEO region

12:00 – 12:20 A. Barea, H. Urrutxua and L. Cadarso – Optimal large-scale object selection and trajectory planning for active space debris removal missions

12:20 – 12:40 E.M. Klein and Y. Yang – Laser/matter interaction modelling for space debris removal

12:40 – 13:00 R. Vilhena de Moraes, J.P.S. Carvalho, C.M. Gonçalves and A.F.B.A. Prado – Orbital evolution of space debris around the Earth: solar sail, an alternative to remove space debris

13:00 – 14:30 Lunch break

14:30 – 18:00 Orbit Determination and Cataloguing Chair: D. Hautesserres

14:30 – 14:50 R. Domínguez, N. Sánchez-Ortiz, P. Quiles and D. Oliviero – Drivers for the cataloguing of space objects

14:50 – 15:10 A. Vananti and T. Schilknecht – Distance between Keplerian orbits in the correlation of short arc radar tracks

15:10 – 15:30 A. Pastor, D. Escobar, M. Sanjurjo-Rivo and A. Águeda – Initial orbit determination methods for track-to-track association

15:30 – 15:50 A. Díez, A. Pastor, S. López, A. Souto, P. García and D. Escobar – Covariance determination from operational orbits and sensor data


16:10 – 16:30 J. Siminski – Mixed-model orbit determination for closely-spaced objects

16:30 – 16:50 G. Baù, H. Ma, D. Bracali Cioci and G. F. Gronchi – An IOD method for LEO objects observed by radar

16:50 – 17:10 L. Pirovano and R. Armellin – Probabilistic data association as intersection of orbit sets

17:10 – 17:30 M.A. Vázquez, J. López-Santiago, M. Sanjurjo-Rivo and J. Míguez – A particle filtering method for the opportunistic update of a space debris catalogue

17:30 – 18:00 Software competition (D. Hautesserres / R. Armellin / J.F. San-Juan)


Thursday 25

9:00 – 09:45 Plenary: Lamberto Dell’Elce – Multi-Phase Averaging of Time-Optimal Low-Thrust Transfers (in collaboration with J.-B.Caillau and J.-B. Pomet).

9:45 – 12:25 Propagation Techniques and Applications Chair: H. Urrutxua

09:45 – 10:05 H. Namazyfard, A.J. Rosengren, D. Amato, C. Bombardelli and L. Dell’Elce – Accurate recovery of ephemerides from TLEs of resonant orbits

10:05 – 10:25 M. Nugnes and C. Colombo – Introduction to a Keplerian-orbital-element-based optimisation approach via differential dynamic programming


10:45 – 11:05 T. Nie and P. Gurfil – On propagation of lunar frozen orbits

11:05 – 11:25 R. Armellin, D. Gondelach, A. Wittig and Y. He – High order transfer map method: theory and applications

11:25 – 11:45 A. Wittig and H. Urrutxua – Differential Algebra ODE solvers for orbit propagation

11:45 – 12:05 P. Bartram, H. Urrutxua and A. Wittig – The performance of multistep collocation methods on problems in Astrodynamics

12:05 – 12:25 J. Hernando-Ayuso, C. Bombardelli and G. Baù – Orbit uncertainty propagation in Dromo variables with a time element


14:00 – 15:40 Orbital Conjunctions Chair: R. Armellin

14:00 – 14:20 D. McKnight, K. Molloy, S. Speaks and C. O’Hara – Machine Learning exercise for MCMA conjunction prediction improvements

14:20 – 14:40 J.L. Gonzalo, C. Colombo and P. Di Lizia – Drag- and SRP-induced effects in uncertainty evolution for close approaches

14:40 – 15:00 M. Romano, C. Colombo and J.M. Sánchez – Line sampling procedure for extensive planetary protection analysis

15:00 – 15:20 N. Reiland, A.J. Rosengren and C. Bombardelli – Assessing collision risk using brute-force numerical simulations

15:20 – 15:40 R. García-Pelayo – Probability of collision between a rectangular cuboid and small debris


16:00 – 18:00 Software Tools Chair: M. Sanjurjo-Rivo

16:00 – 16:15 E.M. Alessi – The ReDSHIFT software tool for passive debris mitigation. The ReDSHIFT Team

16:15 – 16:30 N. Sánchez-Ortiz, R. Domínguez, J. Nomen and S. Pessina – SSA tools and techniques supporting satellite flight dynamics activities

16:30 – 16:45 A. Lozano, R. Sánchez and J.J. Negrete – Application of SAPO, the GMV’s Semi-Analytical Orbit Propagator, in flight dynamics operational tools

16:45 – 17:00 A. Morselli – SINTOP: a numerical orbit propagator for satellite operations

17:00 – 17:15 I. Pérez, R. López, E. Vergara, M. San-Martín, E. Segura and H. Carrillo – HOPE: Hybrid Orbit Propagator Environment


17:15 – 17:30 F. Deleflie, L. Sagnières, A. Petit, D. Hautesserres and M. Capderou – Atmospheric reentry predictions with SATlight/STELA from TLE time series. Application on the Tiangong-1 reentry

17:30 – 17:45 P.J. Cefola, J.F. San-Juan and R. López – Validation of the DSST C/C++ version

17:45 – 18:00 A. Wittig, V. Vanick, R. Penn, J. Castle and J. Tyler – Space AR: augmented and virtual reality devices for Astrodynamics visualization

21:00 Gala Dinner: Herventia restaurant (Logroño, La Rioja).

Friday 26

9:30 – 10:15 Plenary: Guillermo Ortega – Overview and Future of Design for Demise Activities at the European Space Agency

10:15 – 11:35 Analytical and Semi-Analytical Methods Chair: J.F. San-Juan

10:15 – 10:35 N. Miguel and C. Colombo – Averaged attitude and orbit dynamics of a planar Earth-orbiting helio-stable solar sail

10:35 – 10:55 P. Izzo, L. Dell’Elce, P. Gurfil and A.J. Rosengren – Transformation between mean and osculating elements in the presence of lunisolar perturbations

10:55 – 11:15 A.J. Rosengren and M. Lara – Averaging the Legendre polynomials for third-body and zonal harmonics perturbations

11:15 – 11:35 M. Sansottera, A.-S. Libert, U. Locatelli and A. Giorgilli – Long-term evolution of extrasolar systems via normal forms


11:55 – 14:00 Dynamics Modelling and Analysis Chair: F. Deleflie

11:55 – 12:15 E.M. Alessi, C. Colombo and A. Rossi – Phase space description of the dynamics due to the coupled effect of the planetary oblateness and the solar radiation pressure perturbations

12:15 – 12:35 D. Hautesserres – 1st-order secular resonances and L-K mechanism for lunar perturbations in SSTO orbits

12:35 – 12:55 J. Daquin, I. Gkolias and C. Efthymiopoulos – Diving into the 2g + h resonance

12:55 – 13:15 M. Duroyon, J. Daquin and J. Pérez – The integrated autocorrelation function as a dynamical chaos indicator

13:15 – 13:35 A. Zlenko, I. Ryabikova – Representations of the force function of two rigid bodies in canonical variables

13:35 – 13:55 I. Gkolias, M. Lara and C. Colombo – Integrable approximations for the coupled interaction of the zonal and lunisolar perturbations for Earth satellites

13:55 – 14:00 Closing (F. Deleflie / R. Armellin)


19:00 Touristic activity (CATARTE): Wine-tasting activity and blues music

Plenary:

DR. HODEI URRUTXUA


Regularization and error propagation in orbit predictions

Hodei Urrutxua*

Abstract

The increasing needs on accuracy and numerical efficiency of orbital calculations calls for high performing techniques, where regularized formulations can play a significant role. These benefits are tightly linked with the capability of minimizing the intrinsic dynamical instabilities of the orbital motion while preserving the geometric elements of the orbit with great fidelity. This talk will revisit the many benefits that regularization brings to orbit propagation, with special emphasis in stability, accuracy and error propagation. In particular, different sources of numerical error will be summarized and how these influence the numerical integration of orbits. Various techniques will be covered that allow to find bounds and reliable estimates for the propagation error, thus providing for means to assess the reliability of orbital solutions. Error propagation will be characterized for various orbital formulations, and techniques to assess and compare the performance of orbital formulations will be presented. Finally, it will be shown how certain features of regularized formulations could be particularly valuable in highly perturbed orbital environments, enabling to define built‐in estimators to monitor the propagation error and estimate the onset of chaos, thus allowing to bound the timespan where the computed orbital solutions remain reliable.

* European Institute for Aviation Training and Accreditation, Rey Juan Carlos University, Fuenlabrada (Madrid) 28943, Spain

Session 1:

SPACE DEBRIS MODELLING AND REMOVAL


Measuring space occupancy of LEO satellites

Claudio Bombardell i*, Gabriele Falco† , Davide Amato‡ and Aaron Rosengren§

Abstract

Preserving and sustaining the circumterrestrial environment as a valuable resource for future space users has motivated space actors to consider mechanisms to control the increase of debris left in orbit. End‐of‐life (EoL) disposal maneuvers by active or passive deorbiting devices, for instance, are strongly recommended for all newly launched objects in Low Earth Orbit (LEO) and have recently been shown to be feasible in Medium Earth Orbit (MEO)[1] and in some cases as far as geostationary altitudes[2].

These remediation actions will become more and more urgent following the launch of upcoming mega‐constellations of satellites. These satellites will indeed be designed in order to deorbit themselves at the end of their operational lifetime most probably by the use of high‐specific‐impulse electrical propulsion systems. However, a small but not negligible percentage of these spacecraft is expected to fail and stay in orbit as debris. The accumulation of debris in mega‐constellation orbits is especially detrimental not only because it will tend to make collision avoidance maneuvers escalate but also because it will gradually make valuable orbital space not accessible to new communication satellites.

An additional mitigation action that can be implemented at a relatively low cost is to minimize the potential interference of a satellite with the rest of the constellation members by a proper adjustment of its initial orbital elements within the limits by which the satellite functionality is not compromised. To this purpose we introduce the concept of “space occupancy volume” (SOV) as the total volume swept by a satellite as it moves around in its orbital space throughout a relevant time span. Objects with non‐intersecting SOVs cannot collide with each other, which opens the possibility of optimizing the orbital design of upcoming LEO mega‐constellations to drastically reduce or possibly eliminate endogenous conjunctions and consequent collision avoidance maneuvers.

In this paper we derive an algorithm to obtain initial conditions that minimize the SOV of a satellite by placing it in a Minimum Space Occupancy (MiSO) Orbit and investigate the relation between non‐optimized initial conditions and the extent of the resulting SOV.

First we show that, under zonal gravity feld perturbations alone, MiSO orbits are equivalent to classical frozen orbits[3]. The geometrical and dynamical characteristics of these orbits, not fully addressed in the literature, are here studied in details. Next, we consider a more realistic model including tesseral harmonics, atmospheric drag, solar radiation pressure and lunisolar perturbations. Optimized MiSO initial conditions for this general case are obtained numerically and applied to relevant cases in line with future mega‐constellation orbits. The extent of the minimum possible SOV achievable is computed and the stability of the MiSO conditions are assessed. Finally we study the growth of the SOV when departing from MiSO conditions.

* Associate Professor, Space Dynamics Group. Technical University of Madrid (UPM), Madrid, 28040, Spain † Graduate Student, Department of Industrial Engineering. University of Naples Federico II, Naples, 80125, Italy ‡ Postdoctoral Research Associate, Department of Aerospace and Mechanical Engineering. University of Arizona, Tucson. 85721 AZ § Associate Professor, Department of Aerospace and Mechanical Engineering. University of Arizona, Tucson. 85721 AZ


References

[1] Rosengren, A. J., Daquin, J., Tsiganis, K., Alessi, E. M., Deleflie, F., Rossi, A., and Valsecchi, G. B., “Galileo disposal strategy: stability, chaos and predictability”, Monthly Notices of the Royal Astronomical Society, Vol. 464, No. 4, 2016, pp. 4063‐4076.

[2] Gkolias, I. and Colombo, C., “End‐of‐life disposal of geosynchronous satellites”, 68th International Astronautical Congress (IAC 2017), Vol. 6, International Astronautical Federation, IAF, 2017, pp. 3613‐3619.

[3] Cook, G., “Perturbations of near‐circular orbits by the Earth's gravitational potential”, Planetary and Space Science, Vol. 14, No. 5, 1966, pp. 433‐444.


Comparison of several methods used to create a synthetic population of the space debris in the GEO region

Alexis Petit* and Daniel Casanova†

Abstract

Lately, the value of synthetic populations has been highlighted by recent studies based on transport simulation, disease modelling, spatial economics, quantitative geography between others. It is especially important when heterogeneous data play a crucial role in population mixing. Consequently, the Space Situation Awareness (SSA) issue present proper characteristics to apply different synthetic population methods. In particular, we know just a few information about space debris population and need to perform predictions, i.e. know the evolution of the hole population to create different scenarios and evaluate their impact in order to take political decisions in different spatial missions.

In this work, we want to design a space debris model and explore different ways to improve it. To do that, we analyze the nature of the available observational data (and the different assumptions made from them). We introduce a Python package which handle orbit propagator and breakup model (both implemented in Fortran) to generate a synthetic population of space debris. This method is used as a black‐box, taking as inputs the natures, the dates, and the locations of the in‐orbit breakups. The parameters inside the space debris model or the inputs can be fitted. Besides, the created population can be weighted to have a synthetic population with particular constraints. Focusing on the space debris population in the GEO region, we compare the results, the possibility of improvement and the limitation of the different methods.

* IFAC‐CNR, Sesto Fiorentino, Italy, [email protected] † Centro Universitario de la Defensa, Zaragoza, Spain. [email protected]


Optimal large‐scale object selection and trajectory planning for active space debris removal missions

Adrian Barea*, Hodei Urrutxua* and Luis Cadarso*

Abstract

Most objects in low Earth orbit are concentrated in certain privileged orbital regions. A high density of objects in these regions can result in collisions that generate new objects, thus increasing the possibility of subsequent collisions, and potentially leading to a cascade effect that can severely impact future space operations [1]. Further research has concluded that active removal of certain objects is necessary to achieve the stabilisation of the number of resident space objects in the most populated zones [2, 3].

Active removal missions will most likely not target a single object, but instead multiple objects could be deorbited in a single mission. This poses the question of which objects should be removed in a particular mission, and in which order. Branch‐and‐bound‐based techniques have been proposed to solve this problem [4‐7]. However, current approaches in literature decide the objects to be removed prior to the sequence and manoeuvre optimisation or consider a small pool of candidate objects from which a fixed number of objects is selected.

The development of a method that considers a pool of candidate objects that is large enough to be representative of the distribution of the most hazardous objects in LEO region and selects the objects to remove, so that the resulting mission is as impactful as limited amounts of resources make possible, would provide a broader insight for the mission analysis of future active debris removal missions.

This research proposes an innovative framework based on well‐known Operational Research methodology that, given a large set of candidate objects with an associated threat value, selects a subset of these objects to be removed, and defines the trajectory that allows to rendezvous with them in an optimal order, so that the threat value of the removed objects is maximised, while a limit mission duration and a Δv budget are imposed as constraints. The proposed algorithm comprises two different levels: the upper level selects the objects to remove so that threat value is maximised, while the lower level checks the feasibility of time and Δv constraints while determining the mission sequence and trajectory.

The upper level is described by an Integer Programming problem, which selects the most promising subset of objects. On the other hand, the lower level is described by a Mixed Integer Non‐Linear Programming problem, which is broken down in an Integer Programming master problem and a NonLinear Programming subproblem using Benders decomposition [8, 9]. The master problem and the subproblem are iteratively solved to find tight upper and lower bounds to the optimal solution.

The efficiency of this algorithm lies in the fact that the upper level can efficiently select the most promising object sets while the lower level can check the feasibility of selecting such objects without having to reach the convergence of the upper and lower bounds of the Benders decomposition, which is the most challenging part of the problem because it encapsulates the nonlinearities of the orbital dynamics.

* European Institute for Aviation Training and Accreditation, Rey Juan Carlos University, Fuenlabrada (Madrid) 28943, Spain


References

[1] Kessler, D., and Courpalais, B., "Collision Frequency of Artificial Satellites ‐ Creation of a Debris Belt," Journal of Geophysical Research‐Space Physics, Vol. 83, No. NA6, 1978, pp. 2637‐2646.

[2] Liou, J., and Johnson, N.L., "A sensitivity study of the effectiveness of active debris removal in LEO," Acta Astronautica, Vol. 64, No. 2‐3, 2009, pp. 236‐243.

[3] Lewis, H.G., White, A.E., Crowther, R., "Synergy of debris mitigation and removal," Acta Astronautica, Vol. 81, 2012, pp. 62‐68.

[4] Cerf, M., "Multiple Space Debris Collecting Mission‐Debris Selection and Trajectory Optimization,"Journal of Optimization Theory and Applications, Vol. 156, No. 3, 2013, pp. 761‐796.

[5] Madakat, D., Morio, J., and Vanderpooten, D., "Biobjective planning of an active debris removal mission," Acta Astronautica, Vol. 84, 2013, pp. 182‐188.

[6] Olympio, J.T., and Frouvelle, N., "Space debris selection and optimal guidance for removal in the SSO with low‐thrust propulsion," Acta Astronautica, Vol. 99, 2014, pp. 263‐275.

[7] Berend, N., and Olive, X., "Bi‐objective optimization of a multiple‐target active debris removal mission," Acta Astronautica, Vol. 122, 2016, pp. 324‐335.

[8] Benders, J.F., "Partitioning procedures for solving mixed‐variables programming problems,"Numerische Mathematik, Vol. 4, No. 1, 1962, pp. 238‐252.

[9] Geoffrion, A.M., "Generalized Benders decomposition," Journal of Optimization Theory and Applications, Vol. 10, No. 4, 1972, pp. 237‐260.


Laser/matter interaction modelling for space debris removal

Erik Marten Klein*† and Yang Yang†‡

Abstract

Past studies have shown that a laser space debris removal system can be a significant alternative compared to other systems. They assumed the target Resident Space Object (RSO) to be cooperative and neglected thereby issues concerning the shape. Further publications have dealt with the effect of different shapes isolated from earth’s space environment and showed that lateral momentum can be a significant error source due to its dependence on the attitude when it comes to orbit modification via ablation effects of a pulsed laser or photon pressure effects of a continuous wave (CW) laser. This proposal suggests the simulated removal of RSOs including target shape effects, deviations of the ideal Gaussian laser beam due to atmospheric turbulences and varies other parameters including pulsed and CW lasers, ground‐based and space‐borne stations and others such as station power and target geometry. Simulation will be done in MATLAB, modelling the target object as compound of facets where each contributes to the resulting momentum. Using this, accelerations can be calculated and handed over to a dedicated propagator. Propagation will be done in all translational and rotational degrees of freedom. Considering the coupled orbit/attitude dynamics shall hold an answer if laser ablation can be successfully applied as debris removal system.

The discussed objectives and their implementation into the simulation environment MATLAB shall provide a tool to simulate and validate future laser engagements to manoeuvre or de‐orbit space objects to avoid collisions. Ground‐based and space‐borne stations are implemented as there are many proposed systems of both types in the literature. Further laser station configuration parameters, such as wavelength, pulse length and power, can be assessed towards its usability. To fully meet the requirements to be able to appropriately compare mentioned types of laser platforms the propagation through turbulent atmosphere is included, since this influence on the beam shape cannot be neglected.

Acknowledgement: the authors would like to acknowledge the support of the Cooperative

Research Centre for Space Environment Management (SERC Limited) through the Australian

Government’s Cooperative Research Centre Programme. The first author also thanks for the help

from Christopher Kebschull.

Erik Marten Klein is an exchange Master student at RMIT. Corresponding email: erik.klein@tu‐braunschweig.de

Yang Yang is a research fellow at RMIT. Corresponding email: [email protected] or [email protected]

* Institute of Space Systems, Technical University of Braunschweig, Hermann‐Blenk‐Str. 23, 38108 Braunschweig, Germany † SPACE Research Centre, School of Science, RMIT University, 124 La Trobe St, Melbourne Victoria, 3000, Australia. ‡ Space Environment Research Centre Limited, Canberra, Australian Capital Territory, 3001, 2610, Australia


Orbital evolution of space debris around the Earth: solar sail, an alternative to remove space debris

Rodolpho Vilhena de Moraes*, Jean Paulo dos Santos Carvalho†, C. M. Gonçalves* and Antônio Fernando Bertachini de Almeida Prado*

Abstract

Space debris remains a threat to the sustainability of space missions. Due to the amount of space debris in Earth orbit, collisions between operating vehicles and space debris can completely prevent the continuation of space missions, especially if there are collisions between debris, which generate even smaller fragments. These cascading collisions can produce Kessler's syndrome (in low‐altitude orbit). For geostationary orbits (GEO) there is also the risk of collision of deactivated objects with active satellites. It is well known that space debris in the GEO ring can remain there for a long period of time (almost circular orbits). We know that the effect of solar radiation pressure (SRP) in orbits around the Earth is mainly considered from the perspective of bodies characterized by a very high area‐to‐mass ratio (balloon satellites). Here we propose to use a solar sail (with a previously calculated area‐to‐mass ratio), such that the perturbations considered, especially the SRP that is the dominant pertubation, act in the debris causing it to increase its orbital eccentricity in a short period of time. The long period evolution of the space debris is highly affected by the perturbations, helping to remove the debris from the GEO ring, which is a very important region for satellites, for example, of telecommunications. Motivated by the work of Casanova et al. (2015), we developed a model to propagate space debris in GEO ring, considering the effect of J2 due to the non‐sphericity of the Earth, perturbations of the Sun and Moon (single‐averaged model), and the solar radiation pressure (single‐averaged model), initially in first order. The SRP model considered in this work is based on equation (17) of the paper Casanova et al. (2015). Our code has been verified by reproducing Figure 1 of the paper cited above. Several simulations are performed with real objects that are in GEO orbit, especially space debris. We find suitable area‐to‐mass ratio values so that the eccentricity and inclination of the solar sail are strongly modified. The perturbations in the solar sail contribute to pushing the space debris into the Earth's atmosphere. One point that we consider important in the use of the solar sail is the technology that uses a clean and abundant source of energy to remove the space debris of the Earth's orbit in a sustainable way.

Keyword: Solar Sail, Space Debris, Third‐Body Perturbation, Solar Radiation Pressure.

Acknowledgements: the authors wish to express their appreciation for the support provided by grants #406841/2016‐0, 301338/2016‐7, 307724/2017‐4, 308823/2015‐0, 420674/2016‐0 from the National Council for Scientific and Technological Development (CNPq); grants #2016/14665‐2 and 2016/24561‐0 from São Paulo Research Foundation (FAPESP) and the financial support from the Coordination for the Improvement of Higher Education Personnel (CAPES).

* Division of Space Mechanics and Control ‐ INPE, São José dos Campos ‐ SP, Brazil † Universidade Federal do Recôncavo da Bahia‐ UFRB, Centro de Ciência e Tecnología em Energia e Sustentabilidade, Feira de Santana‐BA, Brazil

Session 2:

ORBIT DETERMINATION AND CATALOGUING


Drivers for the cataloguing of space objects

Raul Domínguez*, Noelia Sánchez‐Ortiz*, Pablo Quiles* and Dario Oliviero*

Abstract

This paper presents the main drivers for the design and implementation of the system for cataloguing Earth orbiting objects used by DEIMOS: CORrelation Tool (CORTO) and provides information of real cataloguing activities based on optical observations.

The approach undertaken in CORTO is based on a three step process: first correlation in the basis of comparison of observation with expected visibility considerations, a second orbit determination compatibility cross‐check based on the filtering residuals, and a further manual processing of generated objects is executed to identify failures in the automatic correlation that could lead to duplicated objects.

CORTO cataloguing system is accompanied by a set of auxiliary tools, which enhance the capabilities of the system to ensure the correctness of the cataloguing process. Two modules, CORTOHouseKeeping and CORTOEditor allow maintaining and modifying the CORTO catalogue accounting for the operator feedback.

Additionally, the auxiliary tools, normally used together with CORTO, include CALMA (for calibration of observation stations), OEMCHECK (to evaluate consistency of orbital data) and CHOCO (which allows correlating the observed objects with those in the TLE dataset). This tool serves to assign the international ID to the CORTO objects, but is not mandatory for successful correlation of objects within CORTO. The catalogue is finally made available through a restricted web system (CAWEB) that allows the user to search objects, analyse the resulting accuracy, the evolution of the orbital information computed for each catalogued object, etc.

Together with the description of the cataloguing approach, the presentation provides a summary of the processing of observations from DEIMOS Sky Survey (DeSS) sensors located in Spain. Optical observations are used to feed the cataloguing system CORTO, allowing the creation of a catalogue of high‐altitude objects which are observable from southern Europe. In particular GEO ring longitudes covering Europe are well represented. The achievable accuracy of the observed orbits can reach values around 10‐100 meters. Object manoeuvres can also be observable.

* DEIMOS Space, 28760, Madrid, Spain.


Distance between Keplerian orbits in the correlation of short arc radar tracks.

Alexandro Vananti* and Tomas Schildknecht†

Abstract

Abstract Radar observations are used to track space objects in the Low Earth Orbit region. The observations consist of short arc tracks containing range and angular information. The association of two or more observed tracks of the same space objects is in general necessary to calculate the orbit of the object with a better accuracy. In the radar case the availability of range and angle measurements allows the computation of an initial orbit from a single track. As a consequence the association of tracks can be based on a direct comparison of the calculated initial orbits. In this work a new definition of distance between Keplerian orbits is proposed. The definition extends the existing formulation based on only five Keplerian orbital elements without the anomaly. The modified distance considers an additional term due to the anomaly. The obtained distance can be scaled as a function of the covariance of the orbital elements and can be expressed as Mahalanobis distance. Some application of the distance definition is shown and compared with results obtained using the orbital distance in curvilinear coordinates.

* Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland [email protected] † Astronomical Institute, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland [email protected]


Initial orbit determination methods for track‐to‐track association

Alejandro Pastor*, Diego Escobar*, Manuel Sanjurjo‐Rivo† and Alberto Águeda*

Abstract

Space Surveillance and Tracking (SST) systems are composed by sensors and on‐ground processing infrastructure devoted to generating a catalogue of RSO: a robust automated database that contains information of every detected object. It represents one of the main outcomes of the SST activities and is the key component for the provision of SST products (e.g. high‐risk collisions, upcoming re‐entries, fragmentations). Track‐to‐track correlation is essential for catalogue build‐up, where it is required to detect objects and estimate their orbits with enough accuracy to ensure proper track‐to‐orbit correlation when new tracks are received after the object is detected.

This work presents Initial Orbit Determination (IOD) methods for track‐to‐track correlation, required to obtain a figure of merit to decide whether certain tracks belong to the same object or not. Classical IOD methods, such as Laplace, Gauss, Gooding or Gibbs, do not provide the required performance, not only in terms of accuracy but also in terms of computational cost. Therefore, new methods are proposed and investigated, involving different methodologies such as Lambert’s problem solvers, double‐r iteration, admissibility regions and observation fitting, among others. Furthermore, some of them make use of the fact that most of the objects are describing nearly circular orbits to obtain an initial solution for their iterative processes. Others do not assume near circular orbits and therefore are suitable for eccentric orbits, being the computational cost sacrificed compromised for a less restrictive method (i.e. larger admissibility region).

IOD methods for both radar tracks and optical tracklets are tackled. In the latter case, the challenge resides in the derivation of enough orbital information from a single tracklet (with less attributables than a radar track) to allow the application of filters and complexity reduction techniques. Hence, IOD methods for optical tracklets are far more complex and resource‐consuming than those for radar tracks and classical approaches are not enough.

In general, IOD methods are complex to assess due to the large number of possible combinations and particular cases. There is not a universal IOD method able to provide reliable results under all scenarios, each one has its particular hypotheses that limit its application. A comparison campaign has been performed to evaluate these aspects on both classical and proposed methods on a wide spectrum of orbits, including eccentric ones such as Geostationary Transfer Orbit (GTO). Finally, the suitability of using these IOD solutions as initial solution for a batch Orbit Determination (OD) is investigated.

Keywords: initial orbit determination, track correlation, catalogue build‐up

* GMV, Calle Isaac Newton 11, Tres Cantos, 28670, Spain, [email protected], [email protected], [email protected] † Universidad Carlos III de Madrid, Madrid, Spain, [email protected]


Covariance determination from operational orbits and sensor data

Adrián Díez*, Alejandro Pastor*, Sergi López*, Álvaro Souto*, Pablo García* and Diego Escobar*

Abstract

One of the main challenges in conjunction analysis is to provide a representative probability of collision. However, the reliability of this parameter directly depends on how well modelled the uncertainty on the objects is modelled, i.e. uncertainty realism. Assuming Gaussian processes, the uncertainty in the state of the objects can be represented by their covariance, which can be directly obtained via classical orbit determination given that the measurements are available. Nevertheless, covariance is not usually provided by most ephemerides sources (operational orbits), thus requiring methods to estimate it with no more information than sets of ephemerids.

In this work, a generic covariance estimation method based on purely statistical consistency analysis between subsequent orbital updates is investigated. This method can be applied to any ephemerides source that does not contain uncertainty information, generating an abacus characterising the uncertainty evolution along the propagation time.

The abacus is obtained by comparing the reference ephemerides, assumed to have the lowest error, against propagated ones. When no reference ephemerid is available, the most recent propagated orbit is considered as reference. Orbital differences are computed in the TNW local frame. By considering a sufficient amount of independent orbits, a statistical trend of the evolution of the differences along time arises and a covariance abacus can be generated. However, it is important to detect and filter outliers to avoid polluting the statistical analysis. They come from two different sources: manoeuvre modelling mismatch and correlated orbit determination processes. The first implies operational satellite manoeuvres not properly captured during one orbital update and can be tackled with an upper‐bound threshold on the orbital differences, while the second is related to the fact that common measurements may have been considered on two orbital updates and entails lower‐bound thresholds.

This methodology is applied to a subset of objects extracted from (former JFSCC) Special Perturbations (SP) High‐Accuracy Catalogue (HAC) catalogue and compared against externals sources data. The resulting covariance can be useful for both internal validation and product provision (i.e. high‐risk collisions, upcoming re‐entries, fragmentations). Furthermore, this work can be applied to the problem of improving the covariance realism of orbit determination algorithms when sensor data is available. The estimated covariance can be considered as observations to ingest an optimisation process of the consider parameters so that the corrected covariance adjusts to those provided by this methodology.

Keywords: uncertainty estimation, covariance abacus, uncertainty realism

* GMV, Calle Isaac Newton 11, Tres Cantos, 28670, Spain, [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]


Mixed‐model orbit determination for closely‐spaced objects

Jan Siminski*

Abstract

Observations of closely‐spaced objects, e.g. satellites co‐located in a cluster in the geostationary orbit, cannot always be unambiguously associated to the correct objects. That is, the observations fit well with multiple candidate objects in the catalogue. False association decisions then lead to corrupted object state estimates in the catalogue.

The ambiguities can only be resolved once more information through new observations is available. In this study an approach based on the probabilistic multi‐hypothesis tracking framework will be explained, which allows estimating the most likely assignment of observations to objects at each time step. The method overcomes the typical issue of exponentially growing number of tested hypotheses or measurement combinations by introducing soft association weights that are estimated along with the orbits of the objects.

The combined orbit determination and association problem is then solved iteratively by alternating between association weights updates and orbit improvement steps similarly to expectation‐maximisation optimization. The disadvantage of the approach is that convergence in only guaranteed locally.

The approach will be illustrated with a case study using optical observations of Eutelsat Hot Bird satellites tracked from the Zimmerwald observatory in Switzerland. Additionally, the effect of outliers on the solution is shown and practical methods to reduce the caused effect are suggested.

* IMS/Space Debris Office, ESA/ESOC, Darmstadt, Germany


An IOD method for LEO objects observed by radar

Giulio Baù*, Hélène Ma†, Davide Bracali Cioci‡ and Giovannni F. Gronchi§

Abstract

We present an initial orbit determination method for satellites in Low Earth Orbits (LEO) observed by radar. The method, also called “Infinitesimal Angles Method”, is an extension of the algorithm introduced in Gronchi et al. (2015) to take into account the effect of the Earth oblateness. A single observation gives the topocentric position of the satellite at time t, that is (ρ, α, δ), where ρ is the topocentric distance (also called “range”), α is the right ascension and δ is the declination. A radar track is a set of at least 4 observations. We assume that: the observations are collected at short time intervals, the measurement of the range is accurate while the line of sight direction, which is given by α, δ, is poorly determined. If we refer to the mean epoch of the radar track, we can calculate the range (ρ), range rate (˙ρ) and the angles α, δ by a polynomial fit. From our assumptions it follows that to compute an orbit, one has to know the values of the angular velocities (˙α, ˙δ) and the corrections of the interpolated values of α, δ, here called infinitesimal angles ∆α, ∆δ. The proposed method tries to link together two radar tracks by solving a system of 8 equations in the 8 unknowns (∆α,∆δ, ˙α, ˙δ)i, i = 1,2, where the subscript refers to the mean epoch of the radar tracks. We assume that the perturbation due to the Earth oblateness is represented by the second zonal term of the geopotential and we consider the secular rates of the ascending node, the argument of perigee and the mean anomaly. Then, the system is composed by: the equation of motion projected onto the line of sight at the two mean epochs, a suitable version of Lambert’s equation and the equations representing the evolution of the Keplerian integrals. Gronchi et al. (2015) showed that this technique is able to correct the angular variables when the dynamics is assumed to be purely Keplerian. We have performed some numerical tests with LEO objects in which the new version of the “Infinitesimal Angles Method” is compared with the previous one.

References

G. F. Gronchi, Linda Dimare, D. Bracali Cioci, H. Ma, On the computation of preliminary orbits for space debris with radar observations, MNRAS, 451,1883– 1891 (2015).

* Department of Mathematics, Largo Pontecorvo 5, 56127 Pisa, Italy, [email protected] † Department of Mathematics, Largo Pontecorvo 5, 56127 Pisa, Italy, [email protected] ‡ Space Dynamics Services s.r.l., Via M. Giuntini 63, 56023 Navacchio, Italy, [email protected] § Department of Mathematics, Largo Pontecorvo 5, 56127 Pisa, Italy, [email protected]


Probabilistic data association as intersection of orbit sets

Laura Pirovano* and Roberto Armellin*

Abstract

The problem of determining the state of resident space objects (RSOs) is fundamental to maintain a collision‐free environment in space, predict space events and perform activities. Due to the development of new technologies and the ever‐growing number of RSOs, the number of observations available is increasing by the day. This calls for more efficient methods able to deal with the amount of data produced. Furthermore, when surveying the sky, the shortarc nature of the observations does not allow for precise orbit determination during a single passage of the object over an observing station: being the detections very close in time, little is known about the curvature of the orbit. Thus, for each observation there is more than one orbit that complies with the observation values. The set of admissible solutions corresponding to a single observational arc is here called the Orbit Set. To reduce the uncertainty on the solution and pinpoint the correct orbit associated with the observation, one needs other independent observations of the same object. The main challenge in this, however, is to determine whether two or more observations pertain to the same object, thus whether they are correlated. This is the problem of data association. Current approaches for the data association problem suffer from either high computational effort, due to the point‐wise based algorithms to keep high accuracy on the dynamical model, or low accuracy, due to the use of simplified dynamics to keep the problem (semi‐)analytical and computationally efficient. We propose a novel approach that exploits Differential Algebra (DA) to overcome the issues from literature. DA is a computing technique that uses truncated power series (TPS) instead of numbers to represent variables, thus allowing us to obtain results of complicated functions as high‐order Taylor polynomials. For the data association problem, we can thus obtain the definition of the Orbit Set with respect to the observation uncertainty in an analytical form, without the need for point‐wise sampling, and propagate the polynomial form independently of the dynamics chosen, thus allowing us to keep accurate dynamics. The technique consists of three different steps.

1. Supposing an observation is available at a first epoch, in the first step an initial orbit determination (IOD) algorithm is developed to compute the state of the orbiting body. The solution is obtained as a Taylor polynomial that links the uncertainty in the observation to the state of the orbiting body. In this way, by evaluating the polynomial for admissible values of the observations, one obtains a range of possible solutions for the IOD problem, which is the Orbit Set (OS). The accuracy of the Taylor expansion is managed by the Automatic Domain Splitting (ADS) tool which estimates and controls the truncation error of the polynomial, by creating a mesh of smaller domains.

2. The state computed at the initial epoch is propagated to a second epoch where a new observation is available. The state is then projected onto the second observation domain, to obtain the observations at the second epoch compatible with the propagated state. DA and ADS are also used during the propagation: the former allows the uncertainties to be propagated to the final epoch keeping a Taylor representation, while the latter manages the truncation error. It is to be underlined that the dynamics considered is independent of the method, which means that one can include all perturbations. Thus, thanks to DA one can obtain an analytical function that maps the initial observation domain onto the observation domain at the second epoch, without having to simplify the

* University of Surrey, Guildford, UK


dynamics. The final map allows for a direct search for overlapping regions, thus looking for possible correlation.

3. The correlation probability of two observation sets is computed. This is the probability that two observations belong to the same object. Having an analytical map between different epochs, each propagation is a simple polynomial evaluation. A subset simulation process, based on the generation of Markov Chains, is adopted to compute the correlation probability using the sum of weighted square residuals as performance function to rank the chains. By describing the angular uncertainties in the observation as Gaussian variables, the performance function behaves like a Chi‐square, allowing us to perform statistical tests to determine the probability.

The algorithm was developed in C++ where the DACE library containing the DA routines was available. The observations were simulated to allow for the validation of the algorithm against real orbital parameters of orbiting objects. Furthermore, an analysis on the sensitivity of the IOD algorithm to different observing strategies was performed. A further analysis to handle even shorter observation arcs has been carried out: indeed, when the arc is too short and/or uncertain, a polynomial regression on the observed angles may reduce the uncertainty and still allow for IOD to be computed. Results are shown for real observations from TFRM observatory.


A particle filtering method for the opportunistic update of a space debris catalogue

Manuel A. Vázquez*, Javier López‐Santiago*, Manuel Sanjurjo‐Rivo* and Joaquín Míguez*

Abstract

The increase in the amount of debris orbiting our planet has become a problem for present and future space missions. Space debris is mainly composed of inactive satellites, rocket bodies and pieces of both detached by collisions. The number of spacecraft has grown in a sustained manner since the 1950s and, as a consequence, debris created by fragmentation of large satellites has increased at a very fast rate. Currently, the number of objects resulting from fragmentation duplicates the number of inactive satellites and rockets. Approximately 20,000 pieces of debris larger than a softball (≈10 cm) have been catalogued and it is believed that more than 2 million objects with a diameter larger than 2 cm orbit the Earth. Actively tracking all these objects is a very expensive task in terms of time and resources. A method to propagate uncertainties into the future up to a new programmed or expected observation in a reliable way is needed to compensate the lack of continuous tracking.

In recent years, several attempts have been made at incorporating probabilistic tools for propagating uncertainties in orbital mechanics [1, 5, 4]. Among the latter, particle filters (PFs) [3, 2] stand out because of their flexibility and generality. PFs are sequential Monte Carlo methods used for Bayesian statistical inference in dynamical models. To be specific, they aim at constructing empirical (sample based) approximations of the probability distribution of the dynamic state variables conditional on the available observations. PFs consist of two stages: a prediction step, where the dynamical model is employed to generate candidate state values (often termed particles), and an update step where the observed data are used to weight these particles according to their likelihood. PFs are well suited for nonlinear tracking problems and, in particular, they appear as very promising tools for tracking space debris.

We introduce a PF‐based methodology that allows, on the one hand, to update the orbital parameters of pieces of space debris when they are sparsely observed and, on the other hand, to reveal previously uncatalogued objects orbiting the Earth. The dynamics of each piece is represented in terms of six (time‐varying) orbital elements, namely its angular momentum (h), eccentricity (e), true anomaly (ν), ascending node (Ω), inclination (i) and perigee (ω). The proposed method starts from a catalogue of objects and it takes advantage of opportunistic (and possibly heterogeneous) celestial observations in order to

(a) identify new objects to be added to the catalogue and

(b) to compute and update a probability distribution for the orbital elements (h, e, ν, Ω, i, ω) associated to each catalogued object.

* Universidad Carlos III de Madrid. [email protected], [email protected], [email protected], [email protected]


500 particles 104 particles

FAR 0.048 0.048

MDR 0.085 0.057

Table 1: False alarm rates (FAR) and missed detection rates MDR) for a computer experiment with synthetic data generated from the TLE catalogue.

The scheme is designed to maintain and update a large‐scale catalogue of objects, as the computational algorithms needed are easily parallelisable. We employ efficient PF‐based techniques to perform data association (i.e., to link each celestial observation to a subset of pieces of debris), to detect new objects, and to generate and update empirical probability distributions for the orbital elements of the catalogued objects. The latter enable us to estimate and predict the complete orbit around Earth of an object at any given time. The estimates can be endowed with expected errors of various types (covariances, collision risks, prediction errors, ...) which are easily computed from the empirical distributions generated by PFs.

We have assessed the validity of the method in a set of numerical experiments using both synthetic and real (publicly available) data. As a preview of our numerical results, Table 1 shows the false alarm rate (FAR) and the missed detection rate (MDR) for a computer experiment involving the assimilation of 126 observations synthetically generated from a set of 40 objects (30 catalogued). The attained FAR is ≈ 5% and the MDR is ≈ 8.5% (when using 500 particles in the PFs) but can be reduced to ≈ 5.7% by increasing the computational effort (by generating 10,000 particles per PF).

Keywords: space debris, particle filtering, uncertainty propagation, orbit prediction.

References

[1] Kyle J DeMars, Robert H Bishop, and Moriba K Jan. A splitting Gaussian mixture method for the propagation of uncertainty in orbital mechanics. Spaceflight Mechanics, 140, 2011.

[2] P. M. Djuri´c, J. H. Kotecha, J. Zhang, Y. Huang, T. Ghirmai, M. F. Bugallo, and J. Míguez. Particle filtering. IEEE Signal Processing Magazine, 20(5):19–38, September 2003.

[3] J. S. Liu and R. Chen. Sequential Monte Carlo methods for dynamic systems. Journal of the American Statistical Association, 93(443):1032–1044, September 1998.

[4] Alinda Mashiku, James Garrison, and J Russell Carpenter. Statistical orbit determination using the particle filter for incorporating non‐Gaussian uncertainties. In AIAA/AAS Astrodynamics Specialist Conference, page 5063, 2012.

[5] James Woodburn and Sergei Tanygin. Detection of non‐linearity effects during orbit estimation. Paper AAS, pages 10–239, 2010.

Plenary:

DR. LAMBERTO DELL’ELCE


Multi‐phase averaging of time‐optimal low‐thrust transfers

L. Dell’Elce*, J.‐B. Caillau† and J.‐B. Pomet*

Abstract

An increasing interest in optimal low‐thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the manoeuvrability of small satellites.

The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as manoeuvre planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables.

Solving the control problem via so‐called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition, PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast‐oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is ap‐pealing. In particular, averaging techniques were used since the early space age to gain understanding into the long‐term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low‐thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system.

The talk focuses on time optimal manoeuvres in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast‐oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the adjoint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajectory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well‐known mean‐to‐osculating transformations of uncontrolled motion.(2) What is the impact of orbital perturbations and their frequencies on the controlled trajectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high‐order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence.

The methodology is finally applied to a deorbiting manoeuvre leveraging on solar radiation pressure. The presence of eclipses makes the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve.

* Université Côte Azur & Inria, Sophia Antipolis, France. † Université Côte Azur & CNRS, Sophia Antipolis, France.

Session 3:

PROPAGATION TECHNIQUES AND APPLICATIONS


Accurate recovery of ephemerides from TLEs of resonant orbits

Hossein Namazyfard*, Aaron J. Rosengren*, Davide Amato*, Claudio Bombardell i† and Lamberto Dell'Elce‡

Abstract

The Two‐Line Elements of Earth‐orbiting objects produced by the U.S. Strategic Command (USSTRATCOM) are publicly available, but no information regarding the accuracy or confidence of this data is provided. Considering the chronic and significant lack of publicly available “actionable” observation data, researchers are generally forced to work with these “pseudo” observations as initial conditions in modeling and prediction capabilities. There have been many interesting approaches to the TLE‐based‐prediction problem over the past decade, mainly based on a careful fit of the TLEs within a prescribed time window to obtain a more accurate initial condition and epoch. Propagations based on these TLE‐orbit‐estimation methods have been carried out over month timespans at best; however, reasonably accurate orbit predictions based on TLEs over decadal timespans is unprecedented, especially for objects that are highly sensitive to initial conditions. To this end, we apply a new dynamically‐inclined route to orbit prediction based on TLEs to obtain a statistically accurate initial condition for orbit propagation of resonant orbits. This method is implemented to recover trajectories of two test cases: the Molniya‐like (i.e., highly elliptical, semi‐ synchronous, critically inclined) orbit of Cosmos 862 and the highly eccentric and distant NASA Eccentric Geophysical Observatory (EGO). In the latter case, the TLE data was missing for a span of 30 years after which tracking was resumed. Using the developed methods, TLE data was utilized to successfully obtain more precise launch conditions of EGO that recover its full trajectory. Concerning Cosmos 862, it is promising that even with uncharacterized input data, we can use such dynamics‐based methods to make decently reliable decadal orbital predictions.

* Department of Aerospace and Mechanical Engineering. University of Arizona, Tucson, 85721 AZ † Associate Professor, Space Dynamics Group. Technical University of Madrid (UPM), Madrid, 28040, Spain ‡ INRIA Sophia Antipolis ‐ Méditerranée, Sophia Antipolis cedex, France


Introduction to a Keplerian‐Orbital‐Element‐based optimisation approach via Differential Dynamic Programming

Marco Nugnes* and Camilla Colombo†

Abstract

One of the classic non‐linear constrained optimal control problems in space applications involves the optimisation of low‐thrust trajectories which are based on propulsion systems granting a greater final operational mass thanks to their high specific impulse despite for the higher time of flight.

One of the methods in solving such difficult control problem is Differential Dynamic Programming (DDP), which is based on the identification of optimal feedback control laws by the discretisation of the dynamics and the application of Bellman optimal principle. In a past work [1] a modified DDP algorithm for the optimisation of low‐thrust trajectories was proposed where the problem is discretised in several decision steps, so that the optimisation process requires the solution of a great number of small problems. In [2] a new second‐order algorithm was developed based on DDP, called Hybrid Differential Dynamic Programming, which maps the required derivatives recursively through first‐order and second‐order state transition matrices. More recently in [2] a Stochastic Differential Dynamic Programming was presented where random perturbations enter the dynamics of the problem and their expected values are computed by the unscented transform. In all these works the formulation of the dynamics and of the DDP procedure has always been in Cartesian coordinates and no attempt was made to exploit the orbital elements for such optimisation method.

In this talk an unconstrained low‐thrust trajectory optimisation through a DDP approach based on Keplerian orbital elements is derived. Classic Keplerian motion is used to model the dynamics of the spacecraft with no addiction of orbital perturbations. The aim of the work is to translate the existing DDP procedures into orbital elements used as state variables which present the advantage to have a more physical interpretation of the orbital scenario together with a smooth dynamics, apart from the true anomaly variation. The cost function to be minimised is simply set to be the square of the magnitude of the control thrust. This talk introduces the first step for the definition of a new approach to solve non‐linear constrained optimal control problems which can handle also more complex element‐based dynamics making use of averaged perturbation techniques.

An interplanetary transfer to Mars is used as example to test the proposed method and a comparison of the results between the new algorithm and the classic Cartesian based DDP is carried out to enhance the differences and assess the performance.

References

[1] Colombo, C., Vasile, M., and Radice, G., “Optimal Low‐Thrust Trajectories to Asteroids through an Algorithm based on Differential Dynamic Programming”, Celestial Mechanics and Dynamical Astronomy, 105:75 (2009), pp. 75‐112. https://doi.org/10.1007/s10569‐009‐9224‐3

[2] Lantoine, G., and Russell, R.P., “A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 1: Theory.” J. Optimization Theory and Applications 154 (2012): 418‐442. DOI:10.1007/s10957‐012‐0038‐1

* Department of Aerospace Science, Politecnico di Milano, Milan, Italy, [email protected] † Department of Aerospace Science, Politecnico di Milano, Milan, Italy, [email protected]


[3] Ozaki, N., Campagnola, S., Funase, R., and Yam, C.H., "Stochastic Differential Dynamic Programming with Unscented Transform for Low‐Thrust Trajectory Design", Journal of Guidance, Control, and Dynamics, Vol. 41, No. 2 (2018), pp. 377‐387. https://doi.org/10.2514/1.G002367


On propagation of lunar frozen orbits

Tao Nie* and Pini Gurfil†

Abstract

Abstract Detecting lunar frozen orbits, characterized by constant mean eccentricity and argument of periapsis, is an old problem. Frozen orbits have been previously found using various dynamical models, incorporating the gravitational field of the Moon and the third‐body perturbation exerted by the Earth. Once the frozen orbits are found, the corresponding mean orbital elements must be converted to osculating elements so as to initialize the orbiter position and velocity in the lunar frame. Thus far, however, there has not been an explicit transformation from mean to osculating elements, which includes the zonal harmonic J2, the sectorial harmonic C22, and the Earth third‐body effect. In the current work, we derive the dynamics of a lunar orbiter under the mentioned perturbations, which are shown to be dominant for the evolution of lunar orbits and use von Zeipel’s method to obtain a transformation between mean and osculating elements. Whereas the dynamics of the mean elements do not include C22, and hence do not affect the equilibria leading to frozen orbits, C22 is present in the mean‐to‐osculating transformation, hence affecting the initialization of the physical lunar orbit. Simulations show that by using the newly‐derived transformation, frozen orbits exhibit better behaviour in terms of long‐term stability about the mean values of eccentricity and argument of periapsis, especially for high orbits.

* Harbin Institute of Technology, Research Center of Satellite Technology; email: [email protected] † Technion ‐ Israel Institute of Technology, Faculty of Aerospace Engineering; email: [email protected]


High order transfer map method: theory and applications

Roberto Armell in*, David Gondelach†, Alexander Wittig‡ and Yanchao He§

Abstract

The propagation and Poincaré mapping of perturbed Keplerian motion is a key topic in celestial mechanics and astrodynamics. The high‐order transfer map (HOTM) method enables efficient mapping of perturbed Keplerian orbits using the high‐order Taylor expansion of a Poincaré or stroboscopic map.

In this talk we will first review the HOTM concept and the importance of coordinates selection for improving its performance in terms of accuracy. We will then apply the method to 1) determine the conditions that generate large‐amplitude bounded motion in the full zonal problem and 2) compute repeat‐ground track orbits and perform their station keeping in a high‐fidelity dynamical model.

* Senior Lecturer in Spacecraft Dynamics, University of Surrey, Guildford, UK † Postdoctoral Researcher, MIT, Cambridge, MA, US ‡ Lecturer in Astronautics, University of Southampton, Southampton, UK § Engineer, Orbit Design Group, DFH Satellite Co., Ltd., Beijing, People’s Republic of China


Differential Algebra ODE solvers for orbit propagation

Alexander Wittig* and Hodei Urrutxua†

Abstract

Numerical integration is a fundamental element in the propagation of satellite orbits. Therefore, it is of the uttermost importance to develop numerical integration techniques that are both accurate and efficient, as the integration process has a direct impact on the fidelity of the computed solution. A wealth of integration techniques exists in the literature, yet only few are commonly used in both academia and industry. While generally sufficient, these classical methods can underperform in stringent applications, unlike highly specific, modern integration methods. A variety of such non‐conventional numerical integration algorithms exists, which might offer substantial performance benefits in highly demanding applications.

In practice, it is also often necessary to propagate not just single points but uncertainties in the initial conditions or problem parameters. Several approaches exist for this purpose, among which high order Differential Algebra (DA) techniques have proven highly successful.

Adapting existing numerical integration schemes to run using DA yields automatic high order expansions of the final state in terms of the initial state, time, or system parameters. This DA flow expansion is at the core of almost every DA application in astrodynamics.

Currently, the most commonly adopted integration for these applications is an explicit, adaptive step‐size Runge‐Kutta method common to astrodynamics. These methods are known for their robustness and ease of implementation; however, they are by no means the most computationally efficient. Since the evaluation of DA operations with high order or high number of variables is computationally expensive, DA flow expansion is usually the most time‐consuming component and hence the performance bottle‐neck of DA‐based algorithms.

The goal of the research presented in this talk is the development of a high‐quality open‐source toolbox of high‐performance integration methods for classical and DA applications using the DACE library. We do so by combining state‐of‐the‐art numerical integration methods with DA‐based flow expansion.

Integration schemes implemented for this comparison include Runge‐Kutta type multi‐stage methods, Adams‐Bashforth and Adams‐Moulton multistep methods, and several other integration schemes such as Everhart’s RADAU and the symplectic Leapfrog scheme. Comparisons of the quality of the resulting approximations to the flow expansion from the different methods will be studied. Furthermore, the efficiency of each method is compared. The comparison is carried out by applying each integration method to several typical problems in the field of astrodynamics.

* Astronautics Group, University of Southampton, Southampton SO17 1BJ, UK, [email protected] † European Institute of Aviation Training and Accreditation, Universidad Rey Juan Carlos, Madrid, Spain, [email protected]


The performance of multistep collocation methods on problems in Astrodynamics

Peter Bartram*, Hodei Urrutxua† and Alexander Wittig‡

Abstract

This talk will first introduce and then present the numerical results of a particular class ofintegrator known as Multistep Collocation Methods (MCMs) [3, p. 290] to a series of orbital dynamics test cases designed to characterise the performance of the integrator whilst also high‐ lighting any possible deficiencies. Careful selection of an appropriate numerical integration technique is crucial to maximise the overall performance of any orbital propagator: ensuring that the computational cost is minimised whilst simultaneously ensuring the correct balance between precision and computational effort is maintained.

MCMs, like other techniques such as RADAU or Everhart’s RADAU [2, 6], make use of Radau spacings to maximise performance. They can be viewed as a blend between multistep and implicit Runge‐Kutta (IRK) methods, as they combine the information of both, previous integration steps (k), and additional evaluations of the derivative (s) within the current step. Combination of these two information sources allows for schemes to be created of order 2s+k−2 thereby allowing for a richness of configurations to be achieved not present within any other integration technique and also enabling very high order schemes to be reached for a given computational cost. Each independent k‐s pair yields a unique integration scheme complete with individual stability and performance properties which often prove very competitive compared to traditional integration methods [7]. To highlight this, a selection of configurations have been chosen for presentation to illustrate the differences in performance present for differing k‐s pairs, even for a given order. In addition, MCMs are placed into the broader context of numerical performance through comparison to other integration methods such as RADAU [4], Dormand‐Prince [1]] and Shampine‐Gordon [5].

References

[1] J.R. Dormand and P.J. Prince. A family of embedded Runge‐Kutta formulae. Journal of Computational and Applied Mathematics, 6(1):19‐26, mar 1980.

[2] Edgar Everhart. An effcient Integrator that uses Guass‐Radau spacings. Dynamics of Comets: Their Origin and Evolution, 1985.

[3] Ernst Hairer and Gerhard Wanner. Solving Ordinary Differential Equations II, volume 14. Springer, 1991.

[4] Ernst Hairer and Gerhard Wanner. Stiff differential equations solved by Radau methods. Journal of Computational and Applied Mathematics, 111:93111, 1999.

[5] L. Shampine and M. Gordon. Computer Solutions of Ordinary Differential Equations: The Initial Value Problem. W. H. Freeman, 1975.

[6] Hanno Rein and David S Spiegel. IAS 15 : a fast, adaptive, high‐order integrator for gravitational dynamics accurate to machine precision over a billion orbits. Royal Astronomical Society, 1437:1424‐1437, 2015.

[7] Stefan Schneider. Numerical Experiments with a Multistep Radau Method. BIT Numerical Mathematics, 33(May 1992):332‐350, 1993.

* University of Southampton, Southampton, UK, [email protected] † European Institute for Aviation Training and Accreditation, Universidad Rey Juan Carlos, Madrid, Spain, [email protected] ‡ University of Southampton, Southampton, UK, [email protected]


Orbit uncertainty propagation in Dromo variables with a time element

Javier Hernando‐Ayuso*, Claudio Bombardell i† and Giulio Baù‡

Abstract

Accurate and efficient orbit uncertainty propagation is key for fields like spacecraft navigation, space situational awareness and planetary defense. The most elementary methods to propagate into the future the probability distribution function (pdf) associated to the orbital motion are the slow but accurate Monte Carlo simulations and the fast linear methods, which are only a first approximation of the solution. Between these two, a myriad of methods of varying complexity allow to obtain a more accurate representation of the pdf. Examples of these are State Transition Tensors,1 Differential Algebra,2 Gaussian Mixture Models,3 and surrogate models like Polynomial Chaos Expansions4 or Kriging,5 among others.

One possibility to easily improve the performance of all the methods previously mentioned is to change the representation of the state vector by a more advantageous choice. In particular, if the differential equations that describe the orbital motion in the new variables have a more linear behavior than in the old ones, then we can state that the new state representation partially absorbs the nonlinearities of the orbital motion. The price to be paid for this extended linear regime is the introduction of a set of nonlinear algebraic equations to convert between the new and the old variables, but nonlinear algebraic equations are easier to handle than nonlinear differential equations. The conversion of the pdf between two different sets of variables can be dealt using methods like sampling and estimation of moments, series expansions or other techniques shown in.6 The change of the representation of the state was the approach proposed by Junkins et al. when comparing the propagation of the covariance matrix in Cartesian coordinates and equinoctial elements.7 Several authors have followed a similar approach using curvilinear coordinates to ameliorate the known nonlinearity of the Cartesian coordinates.8–12

In previous works, the Dromo formulation has shown a good performance for orbit uncertainty prop‐agation, with specific examples in Near Earth Asteroids (NEAs)13, 14 and Low Earth Orbit (LEO).15 Dromo is a special perturbation method developed by Peláez et al.16 of the Space Dynamics Group (SDG) of the Technical University of Madrid, Spain. Further improvements were published by different collaborators of the SDG: Urrutxua et al.,17 Baù et al.18–20 and Roa et al.21–24 Dromo employs seven non‐singular orbital elements qi, i = 1, . . . , 7 and a fictitious time σ derived from a second order Sundman transformation. Propagating the orbit uncertainty with the traditional Dromo formulation requires introducing additional cumbersome transformations to obtain the propagated pdf for a constant time τ,22, 23 and to avoid them we select the real time τ as independent variable.

Numerical simulations suggest that for non‐circular orbits, the choice of σ in the state vector introduces nonlinear terms in the equations of motion.15 In this paper, the applicability of a Time element in the Dromo formulation is investigated. In particular, we choose the “constant” time element proposed by Baù et al.19 To this end, the fictitious time σ is replaced in the state vector by a time element q0 that is equivalent to the time of periapsis passage τp plus a drift caused by the perturbations. This drift is easily related to the drift between the fictitious time σ and the true anomaly ν, and represents the effect of the orbital perturbations on the intermediate frame used in the Dromo formulation, which is Hansen‐ideal.17

* ispace Inc. Tokyo, Japan † Associate Professor, Space Dynamics Group. Technical University of Madrid (UPM), Madrid, 28040, Spain ‡ Department of Mathematics, Largo Pontecorvo 5, 56127 Pisa, Italy


A time element can greatly improve the linearity of the motion expressed in Dromo variables as shown in Fig. 1. This figure contains the root mean square error (RMS) of the linear propagation of the uncertainty of the anomaly or time element for a LEO orbit under the lunisolar perturbations and a 2×2 spherical harmonics model for the Earth. We considered equinoctial elements (Eq),25 alternate equinoctial elements (Aeq),26 Dromo, Dromo with a constant time element (DromoCTE) and EDromo.20 Equinoctial elements and alternate equinoctial elements employ mean longitude, Dromo employs a fictitious time related to true anomaly, while the fictitious time in EDromo resembles the eccentric anomaly. The most linear behavior is observed for the alternate equinoctial elements and the Dromo with a constant time element. The Dromo formulation offers some advantages compared to the alternate equinoctial elements. For example, it is possible to easily obtain analytical expressions for the gradient matrices needed to integrate the state transition matrix, and it avoids the singularity for i = 180 deg of the alternate equinoctial elements.

The accuracy of the proposed approach will be presented by numerically evaluating the linearization error for different scenarios, and comparing to formulations available in the literature.

Figure 1: Comparison of root mean square error (RMS) of the angular variables for a LEO orbit for equinoctial elements (Eq), alternate equinoctial elements (Aeq), Dromo, Dromo with a constant time element (DromoCTE) and EDromo.

References

[1] R. S. Park and D. J. Scheeres, “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, 10.2514/1.2017.

[2] M. Valli, R. Armellin, P. Di Lizia, and M. R. Lavagna, “Nonlinear Mapping of Uncertainties in Celestial Mechanics,” Journal of Guidance, Control, and Dynamics, Vol. 36, January 2013, pp. 48– 63, 10.2514/1.58068.

[3] D. Giza, P. Singla, and M. Jah, “An approach for nonlinear uncertainty propagation: Application to orbital mechanics,” AIAA Guidance, Navigation, and Control Conference, No. 2009‐6082, Chicago, Illinois, 10‐13 August 2009, 10.2514/6.2009‐6082.


[4] B. A. Jones, A. Doostan, and G. H. Born, “Nonlinear propagation of orbit uncertainty using non‐ intrusive polynomial chaos,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 2, 2013, pp. 430–444, 10.2514/1.57599.

[5] C. Tardioli, M. Kubicek, M. Vasile, E. Minisci, and A. Riccardi, “Comparison of non‐intrusive approaches to uncertainty propagation in orbital mechanics,” No. AAS 15‐545, San Diego, California, American Astronautical Society, 2015, pp. 3979–3992.

[6] R. M. Weisman, M. Majji, and K. T. Alfriend, “Analytic characterization of measurement uncer‐ tainty and initial orbit determination on orbital element representations,” Celestial Mechanics and Dynamical Astronomy, Vol. 118, Feb 2014, pp. 165–195, 10.1007/s10569‐013‐9529‐0.

[7] J. L. Junkins, M. R. Akella, and K. T. Alfriend, “Non‐Gaussian error propagation in orbital me‐ chanics,” The Journal of the Astronautical Sciences, Vol. 44, October‐December 1996, pp. 541–563.

[8] J. Hernando‐Ayuso and C. Bombardelli, “Orbit covariance propagation via quadratic‐order state transition matrix in curvilinear coordinates,” Celestial Mechanics and Dynamical Astronomy, Vol. 129, Sep 2017, pp. 215–234, 10.1007/s10569‐017‐9773‐9.

[9] V. T. Coppola and S. Tanygin, “Using Bent Ellipsoids to Represent Large Position Covariance in Orbit Propagation,” Journal of Guidance, Control, and Dynamics, Vol. 38, No. 9, 2015, pp. 1775– 1784, 10.2514/1.G001011.

[10] R. G. Melton, “Time‐explicit representation of relative motion between elliptical orbits,” Journal of Guidance, Control, and Dynamics, Vol. 23, No. 4, 2000, pp. 604–610, 10.2514/2.4605.

[11] C. M. Lane and P. Axelrad, “Formation design in eccentric orbits using linearized equations of relative motion,” Journal of Guidance, Control, and Dynamics, Vol. 29, No. 1, 2006, pp. 146–160, 10.2514/1.13173.

[12] K. Hill, K. Alfriend, and C. Sabol, “Covariance‐based uncorrelated track association,” AIAA/AAS Astrodynamics Specialist Conference, No. AIAA 2008‐7211, Honolulu, Hawaii, 18‐21 August 2008.

[13] J. Hernando‐Ayuso and C. Bombardelli, “Orbit Uncertainty Propagation Using Dromo,” AIAA/AAS Astrodynamics Specialist Conference, No. AIAA 2016‐5632, 2016, 10.2514/6.2016‐5632.

[14] J. Hernando‐Ayuso, C. Bombardelli, and G. Bau`, “Uncertainty propagation in the N‐body problem using Dromo elements,” Acta Astronautica, 2017 (in press), 10.1016/j.actaastro.2017.12.030”.

[15] J. Hernando‐Ayuso and C. Bombardelli, “Orbit Uncertainty Propagation around Non‐Spherical Bod‐ ies Using the Dromo Formulation,” 26th International symposium on Space Flight Dynamics, held together the 31st International Symposium on Space Technology and Science, No. 2017‐d‐071, Mat‐ suyama, Japan, 3‐9 June 2017.

[16] J. Peláez, J. M. Hedo, and P. R. d. Andrés, “A special perturbation method in orbital dynamics,” Celestial Mechanics and Dynamical Astronomy, Vol. 97, No. 2, 2007, pp. 131–150, 10.1007/s10569‐ 006‐9056‐3.

[17] H. Urrutxua, M. Sanjurjo‐Rivo, and J. Peláez, “DROMO propagator revisited,” Celestial Mechanics and Dynamical Astronomy, Vol. 124, No. 1, 2015, pp. 1–31, 10.1007/s10569‐015‐9647‐y.

[18] G. Baù, C. Bombardelli, and J. Peláez, “A new set of integrals of motion to propagate the perturbed two‐body problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 116, No. 1, 2013, pp. 53–78, 10.1007/s10569‐013‐9475‐x.

[19] G. Baù and C. Bombardelli, “Time Elements for Enhanced Performance of the Dromo Orbit Propagator,” The Astronomical Journal, Vol. 148, No. 3, 2014, p. 43, 10.1088/0004‐6256/148/3/43.

[20] G. Baù, C. Bombardelli, J. Peláez, and E. Lorenzini, “Non‐singular orbital elements for special per‐ turbations in the two‐body problem,” Monthly Notices of the Royal Astronomical Society, Vol. 454, No. 3, 2015, pp. 2890–2908, 10.1093/mnras/stv2106.

[21] J. Roa and J. Peláez, “Orbit propagation in Minkowskian geometry,” Celestial Mechanics and Dynamical Astronomy, Vol. 123, Sep 2015, pp. 13–43, 10.1007/s10569‐015‐9627‐2.

[22] J. Roa and J. Peláez, “Frozen‐anomaly transformation for the elliptic rendezvous problem,” Celestial Mechanics and Dynamical Astronomy, Vol. 121, No. 1, 2015, pp. 61–81, 10.1007/s10569‐014‐9585‐0.

[23] J. Roa and J. Peláez, “The theory of asynchronous relative motion I: time transformations and nonlinear corrections,” Celestial Mechanics and Dynamical Astronomy, Vol. 127, Mar 2017, pp. 301– 330, 10.1007/s10569‐016‐9728‐6.

[24] J. Roa Vicens and J. Peláez Álvarez, “Efficient trajectory propagation for orbit determination prob‐ lems,” AAS/AIAA Astrodynamics Specialist Conference, No. AAS‐15‐730, Vail, Colorado, USA, Univelt, August 2015.

[25] R. A. Broucke and P. J. Cefola, “On the equinoctial orbit elements,” Celestial Mechanics, Vol. 5, No. 3, 1972, pp. 303–310.

[26] J. T. Horwood, N. D. Aragon, and A. B. Poore, “Gaussian sum filters for space surveillance: theory and simulations,” Journal of Guidance, Control, and Dynamics, Vol. 34, November 2011, pp. 1839 – 51, 10.2514/1.53793.

Session 4:

ORBITAL CONJUNCTIONS


Applying machine learning to enhance conjunction predictions

Darren McKnight*, Kevin Molloy†, Seth Speaks* and Cullen O’Hara*

Abstract

Background

The Massive Collision Monitoring Activity (MCMA) monitors and characterizes the interactions between massive derelicts found in four clusters in low Earth orbit (LEO). These 500 objects comprise over 800,000 kg in four clusters identified by their respective centering altitude: CC775, CC850, CC975, and CC1500. These clusters each contain from 35‐300 intact rocket bodies and non‐operational payloads that were deployed by the rocket bodies (i.e., these object pairs are operationally‐related.

More than two years of monitoring can be summarized in three key observations:

There are encounters on roughly a monthly basis that meet or exceed the probability of collision when Iridium and Cosmos 2251 collided in 2009.

The encounters between these paired massive intact objects greatly exceeds the average encounter rate of the debris population.

Due to the size and altitude of these massive derelicts, predictions performed three days before the event can compute the final miss distance within 6%.

* Integrity Applications Incorporated, Virginia, USA, dmcknight@integrity‐apps.com † James Madison University (JMU), Virginia. USA.


Hypothesis

If the object type (i.e., payload or rocket body), tumble rate, more detailed solar activity, and interactions with the diurnal bulge/dip are considered the ability to predict a conjunction three days out can be improved to 1%.

Each of these four factors had data/models compiled for use in a machine learning experiment.

Object type is well known along with the length, diameter, and mass for each.

Tumble rate measurements are incomplete and inconsistent but a preliminary model was created.

Solar activity values for F10.7 cm solar flux, Ap, Kp, and Dst provided by Baylor University are used.

A diurnal bulge/dip model was created to determine absolute and relative time spent in bulge/dip leading up to every encounter.

Preliminary Results

Classification exercises are completed for quantifying the sensitivity of the four factors on the accuracy of existing conjunctions. Once these attributes have been refined as to their ability to influence conjunction accuracy, they will be combined into a machine learning study.


Drag‐ and SRP‐induced effects in uncertainty evolution for close approaches

Juan Luis Gonzalo*, Camilla Colombo† and Pierluigi Di Lizia‡

Abstract

A key characteristic of drag and solar sails is their high area‐to‐mass ratio, which provides great cost‐efficiency for tasks such as end‐of‐life deorbiting. However, it also negatively affects collision risk by increasing the spacecraft’s collisional Cross‐Sectional Area (CSA), and the stronger perturbing accelerations from drag and Solar Radiation Pressure (SRP) can contribute to a faster growth in position and velocity uncertainties. This second effect is especially relevant when there are additional uncertainties regarding the magnitude of the drag‐ and SRP‐related accelerations, for instance due to uncertainties in attitude which in turn affect the CSA. These considerations are not restricted to sails but apply to any object with a significant area‐to‐mass ratio. Other relevant cases can be rocket upper stages, as their elongated shape and uncontrolled attitude can difficult the reliable estimation of their CSA over time.

In this work we deepen into the effects of drag and SRP in uncertainty evolution, focusing on applications related to close approaches. Numerical and semi‐analytical methods based on planODyn for the propagation of the covariance ellipsoid representing the uncertainties in position and velocity are presented and compared, both in terms of accuracy and computational cost. Although sails are a clear application case the results are not restricted to this kind of objects, and a sensitivity analysis over a significant range of area‐to‐mass ratios is performed. Furthermore, different propagation times and uncertainty levels for area‐to‐mass and the drag and SRP coefficients are considered, to assess their influence. The b‐plane of the nominal close approach is used to project and characterise the evolution of the uncertainties from a physical point of view, relating it to collision risk and the design of collision avoidance manoeuvres. The results show that drag and SRP effects in uncertainty propagation can be neglected for small area‐to‐mass and short timespans but become significant as the area‐to‐mass ratio increases, especially when combined with uncertainties over the effective CSA and the drag and SRP coefficients.

* Postdoctoral research fellow, Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy , Via la Masa 34, 20156, Milan, Italy † Associate professor, Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy , Via la Masa 34, 20156 Milan, Italy ‡ Assistant professor, Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy , Via la Masa 34, 20156 Milan, Italy


Line sampling procedure for extensive planetary protection analysis

Matteo Romano*, Camilla Colombo* and José Manuel Sánchez Pérez†

Abstract

Interplanetary missions must comply to planetary protection requirements, which set limits to the impact probability of spacecraft or launcher upper stages with celestial bodies that may support life (e.g. Mars, Europa, Enceladus). During the design of such missions, uncertainties in the design parameters of the spacecraft, random failures, errors in the determination of its state, propagation errors due to the chaotic n‐body dynamics are analysed to verify that the planetary protection requirements are respected. A conventional verification using Monte Carlo (MC) method is expensive in terms of numerical resources, since the requirements also include high confidence levels of the probability estimates.

The main goal of the proposed approach is to reduce the computational load of the analysis, by employing the Line Sampling (LS) method as an alternative to the MC method for the estimation of the probability of impact between the propagated body and a celestial body. Thanks to its higher efficiency in sampling the initial uncertainty distribution resulting in fewer samples required to obtain the same confidence level, LS offers a method which can be directly used in the mission design, providing additional information about impact regions inside the uncertainty distribution, via the recognition of the time windows where close approaches with planets occur, and repeated LS applications.

The methods described above were implemented in the SNAPPshot tool suite for the verification of the compliance to planetary protection requirements originally developed at the University of Southampton in the framework of an ESA study [1] and now continued at Politecnico di Milano. They will be explained and used together to analyse different interplanetary missions as test cases, such as the ESA/NASA Mars Sample Return mission, and the ESA mission JUICE for the Jovian moons.

References

[1] Letizia F., Colombo C., Van den Eynde J., Armellin R., Jehn R., SNAPPSHOT: Suite for the numerical analysis of planetary protection, 6th International Conference on Astrodynamics Tools and Techniques (ICATT), 14‐17 Mar. 2016, Darmstadt.

*Dept. of Aerospace Science and Technology, Politecnico di Milano, Milano, Italy, [email protected], [email protected] † European Space Operations Centre (ESOC), ESA, Darmstadt, Germany, [email protected]


Assessing collision risk using brute‐force numerical simulations

Nathan Reiland*, Aaron J. Rosengren* and Claudio Bombardelli†

Abstract

With the impending deployment of several thousand telecommunications satellites into low‐Earth orbit (LEO) as components of various mega‐constellations, it is essential that a satellite‐satellite close encounter baseline is established, such that the efficacy of various encounter‐prediction and collision‐probability algorithms may be evaluated. To that end, we develop a new algorithm for predicting collisions among large sets of earth‐orbiting satellites through the use of brute‐force numerical sim‐ulations. The equations of motion are integrated with a sufficiently‐small time step, vis‐à‐vis the relative velocity of the satellites and their trajectories are compared. Drawing inspiration from Hoots et. al. (1984), a sequence of “filters” is implemented, so as to reduce the computational expense of the algorithm. Each subsequent filter consists of integrating the equations of motion with a smaller time step than the previous filter. The desired accuracy is maintained due to the fact that within a certain distance, the motion of two earth‐orbiting satellites can be considered rectilinear. As a result, the time step is selected such that the distance traversed by the satellites at each step is significantly less than the entire interval of rectilinear motion. Thus, only the events where at any given time step, two objects are within the interval of rectilinear motion are further considered. The efficacy of this technique is tested by creating a number of hard‐to‐detect collision events and propa‐gating the trajectories of the involved objects backwards in time to an initial epoch. The brute‐force approach is then applied to the anticipated orbital environment following the deployment of the SpaceX, OneWeb, and Telesat mega‐constellations. This work was performed in the larger context of assessing collisions in satellite mega‐constellations with the ultimate goal of providing operators and researchers with an efficient and effective means for evaluating collision risk during the design process of satellite constellations.

* The University of Arizona, Tucson, United States † Technical University of Madrid, Madrid, Spain


Probability of collision between a rectangular cuboid and small debris

Ricardo García‐Pelayo*

Abstract

The probability of collision between a rectangular cuboid and point‐like debris is computed, under the assumptions of the short‐encounter model. The computation is exact in the sense that it is not based on approximations such as the enveloping sphere approximation, but on a very efficient algorithm to compute the integral of Gaussian over the projection of a rectangular cuboid on the collision plane.

*UPM‐Universidad Politécnica de Madrid, Spain.

Session 5:

SOFTWARE TOOLS


The ReDSHIFT software tool for passive debris mitigation

The ReDSHIFT Team (Elisa Maria Alessi*)

Abstract

The “Revolutionary Design of Spacecraft through Holistic Integration of Future Technologies" is a project funded by the European Commission Horizon 2020, Framework Programme for Research and Innovation (2014‐2020), under the topic “PROTEC‐1‐2015: Passive means to reduce the impact of Space Debris".

ReDSHIFT has tackled the passive mitigation of the space debris problem, by applying advanced astrodynamics tools and simulation suites to accurately depict the circumterrestrial environment in terms of long‐term dynamics, and by investigating novel design‐for‐demise processes and shielding techniques. In particular, the project has assessed the future evolution of the population, also considering the improved disposal guidelines arisen from the dynamical mapping obtained, and has prototyped 3D‐printed cubesats and components and tested them against hypervelocity impacts and atmospheric reentry.

One of the main output of project is the ReDSHIFT software tool which encompasses most of the results obtained. It summarizes the holistic vision of project, and it is targeted to satellite manufacturers, space mission designers, space agencies and researchers. The tool is conceived to provide and evaluate the disposal strategies that are available to the space operator at the end‐of‐life of a given Earth mission. The evaluation is done by considering the required cost of the possible disposal trajectories in terms of impulsive ∆v−budget or area of a solar and drag sail, in terms of collision risk during the disposal trajectory and in terms of percentage of demise in case of reentry. The tool is designed under the OpenSF architecture.

In the talk, we will describe in detail the modules composing the software and show practical examples.

* IFAC‐CNR, Via Madonna del Piano 10, I‐50019 Sesto Fiorentino (FI), Italy


SSA tools and techniques supporting satellite flight dynamics activities

Noelia Sánchez‐Ortiz*, Raul Domínguez*, Jaime Nomen* and Stefano Pessina†

Abstract

SSA orbital determination is based on the processing of radar, telescope and recently SLR observations. All these kinds of observations are based on the lack of collaboration from the object. Contrary to the case of SSA approaches, orbit determination for satellite flight dynamics operations is normally based on collaborative range measurements from a network of stations.

Since June 2018 EUMETSAT has incorporated optical observations into the operational orbit determination process for the 4 geosynchronous satellites METEOSAT, with the objective of assessing the suitability of optical data to support the operational orbit determination, including the calibration of the satellite manoeuvres. In this process, it is evaluated the quality of the rangeonly, optical‐only, and range‐optical data fusion approaches. The processing of the measurement data is being performed by two separate teams at Eumetsat and Deimos. The optical measurements are provided currently by the Deimos Sky Survey (DeSS) telescopes, using additional sensors as backup in case of adverse weather conditions or technical issues.

This paper presents the observation profile, the orbit determination approaches, using a batch least square with a two‐weeks rolling window, identifying the orbital quality and capability to estimate manoeuvres executed by the satellites, all based on tools and techniques from SSA field.

* DEIMOS Space, SLU, Spain. † European Organization for the Exploitation of Meteorological Satellites, EUMETSAT


A look into the numerical performance of the DROMO family of propagators

Jesús Peláez*, Virginia Raposo‐Pulido* and Hodei Urrutxua†,

Abstract

DROMO is an orbital propagator developed in 2000 by the Space Dynamics Group at the Technical University of Madrid (Peláez, 2003; Peláez, 2007). This special perturbation method is characterized by a system of 8 differential equations of motion, comprised by 7 non‐classical orbital elements and the physical time, as a function of a fictitious time defined upon a Sundman‐type time‐regularization. The DROMO formulation is tightly linked to the osculating orbital plane and naturally stems from the "ideal frame" concept introduced by Hansen (Hansen, 1857). As such, it is also related to other regularized formulations that also rely on Hansen's ideal frame, e.g. (Deprit, 1975; Palacios, 1996) and therefore shows some similarities with the aforementioned formulations.

Over the last decade, different variations and modifications of the DROMO formulation have been proposed, developed and studied. These include reformulations of the original method (Baù, 2013; Urrutxua, 2016), the inclusion of time‐elements in lieu of the physical time (Baù, 2014), or special versions tailored for elliptical orbits (Baù, 2014a) as well as for hyperbolic orbits (Roa, 2015; Roa, 2015a). Recently, yet another new reformulation of DROMO has been developed, which is particularly well‐suited for highly perturbed environments (Raposo‐Pulido, 2019). All these versions of DROMO show distinct features and advantages for given orbital scenarios, as well as an intrinsically particular efficiency; also, they exhibit a very different performance depending on the numerical integration scheme that is used.

This talk will first provide a comprehensive overview of the aforementioned variants of the DROMO family of regularized orbital formulations, their characteristic features, and the orbital scenarios that they have been specifically design for. Next, by means of a carefully chosen set of example scenarios, their numerical performance will be briefly assessed for a set of different numerical integration schemes, in order to highlight the dependence with the integrator, as well as to investigate the specific integrators that allow to enhance the performance of each DROMO formulation.

References

[1] Peláez, J. & Hedo, J. M.: "Un método de Perturbaciones Especiales en Dinámica de Tethers", Monografías de la Real Academia de Ciencias de Zaragoza, 2003, 22, 119‐140.

[2] Peláez, J., Hedo, J. M. & de Andrés, P. R.: "A Special Perturbation Method in Orbital Dynamics", Celestial Mechanics and Dynamical Astronomy, 2007, 97, 131‐150.

[3] Roa, J. & Peláez, J.: “Orbit propagation in Minkowskian geometry”, Celestial Mechanics and Dynamical Astronomy, 2015, Vol. 123, 13–43.

[4] Hansen, P. A.: "Auseinandersetzung einer Zweckmässigen Method zur Berechnung der absoluten Störungen der Kleinen Planeten", Abh. der Math.‐Phys. Cl. der Kon. Sachs. Ges. Der Wissensch., 1857, 5, 41‐218.

[5] Deprit, A.: "Ideal Elements for Perturbed Keplerian Motions", Journal of Research of the National Bureau of Standards ‐‐ B. Mathematical Sciences, 1975, 79B, 1‐15.

[6] Palacios, M. & Calvo, C.: "Ideal frames and regularization in numerical orbit computation", Journal of the Astronautical Sciences, AMER ASTRONAUTICAL SOC, 1996, 44, 63‐77.

* Space Dynamics Group, Technical University of Madrid † European Institute for Aviation Training and Accreditation, Universidad Rey Juan Carlos


[7] Baù, G., Bombardelli, C. & Peláez, J.: "A new set of integrals of motion to propagate the perturbed two‐body problem", Celestial Mechanics and Dynamical Astronomy, 2013, 116, 53‐78.

[8] Urrutxua, H., Sanjurjo‐Rivo, M. & Peláez, J.: "DROMO propagator revisited", Celestial Mechanics and Dynamical Astronomy, 2016, 124, 1‐31.

[9] Baù, G. & Bombardelli, C.: "Time Elements for Enhanced Performance of the DROMO Orbit Propagator", The Astronomical Journal, IOP Publishing, 2014, 148, 43.

[10] Baù, G.; Urrutxua, H. & Peláez, J. EDROMO: "An Accurate Propagator for Elliptical Orbits in the Perturbed Two‐Body Problem", Advances in the Astronautical Sciences, 2014.

[11] Roa, J., Sanjurjo‐Rivo, M. & Peláez, J.: "Singularities in Dromo formulation. Analysis of deep flybys", Advances in Space Research, 2015.

[12] Raposo‐Pulido, V., Urrutxua, H. & Peláez, J.: "Dromo Propagator for Highly Perturbed Problems", 29th AAS‐AIAA Space Flight Mechanics Meetings, Maui (Hawaii), 2019. Paper AAS 19‐310.


Application of SAPO, the GMV’s Semi‐Analitical Orbit Propagator, in flight dynamics operational tools

Antonio Lozano*, Roberto Sánchez* and Juan José Negrete*

Semi‐analytical propagators have been identified as a very suitable alternative to numerical and analytical methods, as they allow a remarkable trade‐off between computational cost and accuracy. The formulation is ideal for propagating large catalogues of objects, although its great potential for improving system performances can be extended to other flight dynamics applications, such as end‐of‐life analysis of a satellite, which entails very long propagations, or trajectory optimisation, where a single iteration implies many propagations.

For these purposes, and as part of its internal R&D activities, GMV has developed Semi‐Analytical Propagator for Operations (SAPO), whose formulation is based on the Draper Semi‐analytical Satellite Theory (DSST). SAPO is fully integrated with GMV’s flight dynamics libraries which are used by fully‐operational applications. The software has already completed its first development stage providing an efficient and reliable propagation procedure which includes a widespread environmental perturbations model and satellite orbital manoeuvres.

This work presents the latest improvements of SAPO since the first functional version and highlights the advantages of using a semi‐analytical propagator in commercial tools for real operations. Two software solutions have already been developed at GMV integrating SAPO, and a third one is in progress. The first one, matool (part of GMV’s focussuite family of products), is a long‐term mission analysis commercial tool for geostationary (GEO) station keeping strategies, requiring very long propagations and for which mean orbital elements are more suitable than osculating elements. The second tool, gamo, is a new generic manoeuvre optimiser fully mission‐independent, where the computational cost to compute each iteration is a key factor to meet the end user requirements. Finally, initguess is introduced, a new operational software under development for the Launch and Early Orbit phase (LEOP) manoeuvring strategy initialisation. It will be used for Galileo constellation new satellites starting from launch number 11 foreseen during 2020.

GMV is the main provider of operational flight dynamics solutions in Europe and one of the main references worldwide. In the field of space debris, as part of flight dynamics applications, GMV is a leading company in Space Situational Awareness (SSA) and Space Surveillance and Tracking (SST) activities and is actively participating in the development of the state‐of‐the art of applicable technologies.

* GMV, Isaac Newton 11, 28760 Tres Cantos, Spain. e‐mail: [email protected], [email protected], [email protected]


HOPE: Hybrid Orbit Propagator Environment

Iván Pérez*†, Rosario López**, Eliseo Vergara‡*, Montserrat San‐Martín§*, Edna Segura** and Hans Carril lo**

Abstract

The hybrid methodology allows increasing the accuracy of any type of orbit propagators by predicting the unmodeled effects. In this work, we present an easy‐to‐use environment specially designed for applying the hybrid methodology based on Machine Learning techniques, which we have called Hybrid Orbit Propagator Environment (HOPE). HOPE, which has been created as an R project, uses the H2O framework as the machine‐learning tool. H2O is an open source library which provides an API for the R language and implements many popular machine learning algorithms. HOPE allows data manipulation, applied machine learning algorithms, validation of the models, result visualization and automatically‐generated reports of all the trained models.

* University of La Rioja, 26006 Logroño, Spain † Scientific Computing Group (GRUCACI) ‡ University of Oviedo, 33003 Gijón, Spain § University of Granada, 52005 Melilla, Spain


SINTOP: a numerical orbit propagator for satellite operations

Alessandro Morselli*

Abstract

Accurate numerical propagation plays a key role in the scope of satellite operations. It is indeed necessary to have a reliable, and fast numerical propagator that, besides the nominal mission operations, could support the mission preparation phase, produce trajectories for the computation of orbital events (eclipses, station or lander visibilities, etc.) or provide a reference orbit for nominal and critical operation simulations.

The numerical propagator SINTOP has been developed by Flight Dynamics Test and Validation group of the European Space Operations Centre. A distinctive trait of this propagator is the capability of integrating the spacecraft mass, which is critical in the presence of orbital manoeuvres and can affect the long‐term propagations. In the last decade it was successfully employed to support a wide range of missions, ranging from Earth observations satellites to interplanetary probes like Rosetta. More recently it has been used for validation and verification of trajectories during the aerobraking phase of Exomars and for the check BepiColombo trajectories, including solar‐electric propulsions arcs.

In this work we will provide an overview of the software tool including (1) the available force models such as gravitational models for planets or asteroids/comets, solar radiation pressure models, atmospheric drag, (2) the types of orbital manoeuvres and the associated stop criteria, (3) the propagation arc management, and (4) the input and output file format. Some propagation examples will also be provided: these will include a trajectory around a small body, an aerobraking trajectory, an interplanetary trajectory with electric propulsion, and a propagation for a low‐Earth orbit satellite. Finally a brief description of the preliminary design for the next version of the numerical propagator will be given, focusing mainly on the requirements coming from satellite operations.

* European Space Operations Centre (ESOC), ESA, Darmstadt, Germany, [email protected]


Validation of the DSST C/C++ version

Paul J. Cefola*, Juan F. San‐Juan† and Rosario López

Abstract

The original Fortran‐77 version of the Draper Semi‐analytical Satellite Theory (DSST) exists both as part of the Goddard Trajectory Determination System (GTDS) suite and as the DSST Standalone program. In the 2012 to the 2015 timeframe, the DSST was re‐implemented in Java, in the Orekit flight dynamics library. More recently, the original Fortran‐77 DSST Standalone code has been migrated to C/C++ at the University of La Rioja. For the latter, we have committed to be faithful to the original design as much as possible, with the aim of taking full advantage of the extensive testing against independent data sources. This paper presents results from the validation and verification process of the DSST C/C++ version against the original Fortran‐77 DSST. The resolution of the incompatibilities discovered during this process as well as the issue of the portability of the physical model binary files are two important advances that will be applicable in following iterations of this project.

* Adjunct Professor, Department of Mechanical and Aerospace Engineering, State University of New York at Buffalo, Amherst, NY, USA; Consultant in Aerospace Systems, Spaceflight Mechanics, and Astrodynamics, Vineyard Haven, MA, USA † Scientific Computing Group (GRUCACI). University of La Rioja, 26006 Logroño, Spain


Atmospheric reentry predictions with SATlight/STELA from TLE time series. Application on the Tiangong‐1 reentry.

Florent Deleflie*, Luc Sagnières†, Alexis Petit‡, Denis Hautesserres§ and Michel Capderou**

Abstract

The long term evolution of orbital parameters of a trajectory is efficiently driven following semi‐analytical approaches. Software such as STELA and in fortran prototype SATlight designed to check the compliance of storage orbits with the IADC guidelines propagate time series of mean orbital elements very quickly, with an accuracy level that does not significantly suffer from the averaging approach. By propagating the mean variational equations jointly to the equations of motion, it is even convenient to adjust the mean model to data sets, such as TLE times series, even over long or very long time scales.

We deal in this paper to the atmospheric reentry of the chinese station Tiangong that we analyse a posteriori following this approach. We analyse the impact of the number of TLE, their spread, and the latency before the event, through some charts to be seen as abacus.

We focus also on the ballistic coefficient and the area‐to‐mass ratio that have to be evaluated jointly to the initial (mean) state vector of the satellite. The dynamical modelling is the full model available in SATlight for the motion of the center of mass, as well as, for the first time in SATlight, some equations standing for the spin evolution of the space station, that are integrated jointly to the equations of motion.

* IMCCE / Observatoire de Paris / GRGS, France † McGill University, Montreal, and IMCCE ‡ IMCCE / Observatoire de Paris § CNES, Toulouse, France ** LMD, École Polytechnique, France


Space AR: augmented and virtual reality devices for Astrodynamics visualization

Alexander Wittig*, Vanissa Vanick*, Rob Penn*, Jessica Castle* and Jack Tyler*

Abstract

In this work we present our recent research into the application of virtual and augmented reality devices to the visualization of orbital evolution both for re‐search purposes and well as for teaching and outreach carried out on a UK Space Agency Pathfinder grant.

Virtual reality (VR) devices, such as the Oculus Rift, are rapidly becoming commonplace. Besides computer games, also media such as TV broadcasts are beginning to be delivered in VR. Augmented reality devices (AR), such as the Microsoft HoloLens, are still in an earlier phase of development. But with the recent release of the HoloLens 2 in February of this year, also this technology is being made available to a wider audience.

While VR devices are mostly targeted at entertainment and gaming audi‐ences, Microsoft has significantly changed its focus for the HoloLens 2 towards professional adoption in a variety of fields. Some previous studies have explored possible applications of this technology in the space field, in particular as a supporting device for astronauts [1] and for remote collaborative engineering [2].

In this work, we explore this emerging technology to enable visualisations of astrodynamics related data. Of particular interest is the superior ability of VR/AR technology to convey depth of view to a user compared to traditional flat 2D projections. We will introduce the current state of the art in the field of virtual reality and augmented reality devices, and show some previous ap‐plications of the technology in visualizing complicated high‐dimensional data in other scientific fields. One of the biggest challenges of deploying visualizations on these platforms is the lack of suitable support from common tools used in the scientific community, such as Matlab or Python/Matplotlib. Existing scien‐tific visualizations are hand‐crafted programs, developed in specific 3D gaming engines such as Unity 3D, and for a very particular data set.

Our focus in developing a tool for visualizing in particular orbital evolution data was to design a more generic tool. It is relatively simple for skilled 3D developers to code a specific program to visualize e.g. the solar system in an appealing way. However, such visualizations are custom made for one single purpose. Our visualization tool instead is intended to be easily usable with any existing orbital mechanics codes by simply passing orbital data to the visual‐ization tool through a defined interface. This is achieved through simple JSON formatted files containing ephemeris data and further information on the display properties of an object. The display of this data is then handled by the tool, freeing the user from having to worry about the rendering on and interaction with any particular VR/AR environment. To demonstrate the universal appli‐cability of this tool, we have created simple demonstrators for the solar system and low‐Earth satellites (similar to the stuffin.space website).

We will also bring along several VR devices and a Microsoft HoloLens for in‐terested conference participants to try out for themselves with our little example programs.

* Astronautics Group, University of Southampton, SO17 1BJ, UK


References

[1] “Augmented Reality for AIT, AIV and Operations”, VTT Research, ESA 4000113373/15/NL/MH, June 2017 https://www.esa.int/Our_Activities/Space_Engineering_Technology/Shaping_the_Future/Augmented_Reality_for_AI_AIV_and_Operations

[2] “NASA: Microsoft HoloLens is helping us find the best sites for bases on Mars”, Microsoft News, 2017 https://news.microsoft.com/en‐gb/2017/07/10/nasa‐microsoft‐hololens‐is‐helping‐us‐find‐thebest‐sites‐for‐bases‐on‐mars/

Plenary:

DR. GUILLERMO ORTEGA


Overview and future of design for demise activities at the European Space Agency

Guillermo Ortega Hernando*

Abstract

Spacecraft “Design for Demise” or D4D is currently the solution proposed at system design level to ensure compliance to the risk requirement using space uncontrolled Earth entry. To minimise the risk to human population, a requirement is imposed on spacecraft which will re‐enter that the risk of casualties must be below 10‐4. Compliance with this requirement can be achieved by performing controlled de‐orbit, but the impact in terms of mass and cost can be high. An alternative is to ensure the passive and safe re‐entry within the 25‐year time frame using D4D techniques and technologies. This talk will provide an overview of the current and upcoming activities in the fields surrounding D4D techniques and technologies, the field of orbit propagation that have high relevance for the future of space missions.

* European Space Agency ESTEC, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands

Session 6:

ANALYTICAL AND SEMI‐ANALYTICAL METHODS


Averaged attitude and orbit dynamics of a planar Earth‐orbiting helio‐stable solar sail

Narcis Miguel* and Camilla Colombo*

Abstract

The attitude and orbit dynamics of an Earth‐orbiting helio‐stable solar sail is studied. The class of spacecraft under consideration consist of a bus attached to an adapted Quasi‐Rhombic‐Pyramid (QRP) structure to planar motion whose geometry contains the aperture angle of the panels and the center of mass ‐ center of pressure offset as physical parameters.

The original papers regarding QRP sails justified the stability of the sun‐pointing direction for a fixed center of mass ‐ center of pressure offset value. This result was extended to a larger set of structures, and explicit necessary conditions for stability that relate the two physical parameters mentioned above were also provided in a recent contribution of the authors of this talk. Moreover, non‐neglectable effective stability regions of the attitude phase space close to the sun‐pointing direction were detected. These mainly consist of oscillatory motion.

This talk addresses the coupled attitude and orbit propagation of such spacecraft. The equations of motion are arranged as a fast‐slow dynamical system whose fast subsystem appears to have Hamiltonian structure. The averaging of the small and fast oscillations around the sun‐pointing direction is discussed. The obtained averaged equations of motion have only one time scale and can be interpreted as the equations of motion of a solar sail with a single flat panel with a fixed attitude. The rigorous theoretical analysis is compared with numerical simulations.

* Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy


Transformation between mean and osculating elements in the presence of lunisolar perturbations

Paolo Izzo*†, Lamberto Dell’Elce‡, Pini Gurfil* and Aaron J. Rosengren§

Abstract

Using the direct approach of Kozai [1], we previously derived an analytical short‐period correction for the dominant oblateness perturbation (J2) and compared our results to the classical BrouwerLyddane transformation, while eliminating the critical‐inclination singularity present in the latter treatment. We also provided a numerical algorithm based on the Fast Fourier Transform, akin to Ely [2], for further validation. For highly eccentric and distant satellites, however, the lunisolar effects becomes increasingly relevant and they must be included in any mean‐to‐osculating mapping. Kozai [3] first treated the short‐periodic perturbations due to the moon and the sun, which was then generalized by Fisher [4] for arbitrary obliquity and lunar ecliptic inclination using a canonical formalism. Such short‐period effects were found recently by Luo et al. [5] to be of significance in the long‐term evolution of the Lidov‐Kozai cycles, who also derived an analytical correction for the doubly‐averaged equations. We present herein the short‐period corrections for both the singly‐ and doubly‐averaged theories. Similar to Luo and colleagues, our approach is based on the Milankovitch vectorial elements, which is free of the mathematical singularities that plague other orbital element formulations. Furthermore, we extend our numerical development to include these additional gravitational attractions, providing a first‐order, singularity‐free, fully numerical transformation. This work was performed in the larger context of passive debris removal using resonances and instabilities, where detailed and extensive cartographic stability maps have been produced based on semi‐analytical methods. Yet, without a proper mean‐to‐osculating transformation, it is not clear where in the osculating space one should actually target to place the satellite on a natural disposal trajectory. We correct this often overlooked deficiency herein.

References

[1] Kozai, Y. “The Motion of a Close Earth Satellite.” The Astronomical Journal, Vol. 64, pp. 367–377, 1959.

[2] Ely, T. A. “Transforming Mean and Osculating Elements Using Numerical Methods.” The Journal of the Astronautical Sciences, Vol. 62, No. 1, pp. 21–43, mar 2015. doi:10.1007/s40295‐015‐0036‐2.

[3] Kozai, Y. “Lunisolar Perturbation with Short Period Terms.” Smithsonian Astrophysical Observatory, 1966.

[4] Fisher, D. “Analytic Short Period Lunar And Solar Perturbations On Artificial Satellites.” Celestial Mechanics, Vol. 6, pp. 447–467, 1972.

[5] Luo, D., Katz, B., and Dong, S. “Double‐averaging can fail to characterize the long‐term evolution of Lidov‐Kozai Cycles and derivation of an analytical correction.” Monthly Notices of the Royal Astronomical Society, Vol. 458, pp. 3060–3074, 2016.

* Technion‐Israel Institute of Technology, Haifa, Israel † University of Rome “La Sapienza”, Rome, Italy ‡ INRIA, Sophia Antipolis Mediterranean, France § The University of Arizona, Tucson, United States


Averaging the Legendre polynomials for third‐body and zonal harmonics perturbations

Aaron J. Rosengren* and Martin Lara†

Abstract

Being one of the oldest and most developed topics of investigation in celestial mechanics, averaging provides a particularly effective method for the approximate evaluation of the dy‐ namics of a system. We present here a systematic approach for computing the mean values of general functions encountered in elliptic motion, and apply this to averaging the Legendre ex‐ pansion of various gravitational potentials to higher order. We show that all orbital averages occurring in perturbed Keplerian motion (assuming non‐commensurate orbital frequencies) reduce to averages of certain powers of the orbit radius, which reduce to expressions involving Hansen’s fundamental coefficients. We develop non‐singular secular equations for the evo‐ lution of the orbit angular momentum and eccentricity vectors (Milankovitch elements) for higher‐order third‐body and zonal harmonics perturbations, and study their effects on highly elliptical orbits in circumterrestrial space.

* The University of Arizona, Tucson, United States † Scientific Computing Group (GRUCACI). University of La Rioja, Logrono, Spain


Integrable approximations for the coupled interaction of the zonal and lunisolar perturbations for Earth satellites

Ioannis Gkolias*, Martin Lara† and Camilla Colombo‡

Abstract

The problem of the long‐term evolution of artificial Earth satellites was one of the first challenges that scientists and engineers had to deal with during the Space Era. Due to the lack of computational power, sophisticated analytical techniques were employed to distil valuable information out of the Hamiltonian models. On the other hand, in modern astrodynamics there exist the tools and computational capabilities to propagate billions of orbits over centurial timescales. Any orbital dynamics problem can be tackled by cartographical studies and Monte Carlo simulations.

Despite the usefulness of those methods they lack the insight into the underlying dynamics. This gap can be filled by studying and understanding suitable simplified analytical models. The solution of the main problem of the artificial satellite theory is known since the work of Brouwer, while the solution to the secular third‐body dynamics was given by Lidov and Kozai. However, a complete solution of the coupled interaction for Earth satellites is still lacking. One of the problems that appears in analytical studies is that of selecting a proper reference frame for the reduction. The short‐come appears due to the competing effect of the zonal and third‐body perturbation to drive the orbital plane precession about the equator and the ecliptic respectively. For close Earth satellites the Earth’s oblateness dominates, therefore an equatorial description benefits from the accurate description of the zonal dynamics. On the contrary, for distant Earth satellites the effects of the Sun and the Moon dominate suggesting that the ecliptic plane could serve as an alternative choice, since it provides a better description of the respective third‐body dynamics.

In this work two main approaches will be used for our analytical investigation: a) a derivation of the J2 and third‐body problem with the force model expressed with respect to the ecliptic and b) a classical equatorial derivation of the third body potential coupled with perturbations stemming from higher order zonal harmonics. Some further manipulation allows us to isolate interesting terms that dictate the long‐term evolution. The models are then used in two cases of interest, namely, to study the long‐term stability of distant Earth satellites and to investigate the evolution of a satellite placed in the critical inclination resonance. The computed integrable approximations are compared against higher fidelity models to verify their predictive capabilities. Although in many cases the 1 degree‐of‐freedom reduction is not suitable to capture the complex dynamical evolution due to the gravitational interactions, it can still provide valuable dynamical insight and serve as a starting point for a more refined orbital design and analysis.

* Politecnico di Milano, Milan, Italy. email: [email protected] † Scientific Computing Group (GRUCACI). University of La Rioja, Logrono, Spain. [email protected] ‡ Associate professor, Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy. [email protected]

Session 7:

DYNAMICS MODELLING AND

ANALYSIS

Phase space description of the dynamics due to the coupled effect of the planetary oblateness and the solar radiation pressure perturbations

Elisa Maria Alessi*, Camilla Colombo† and Alessandro Rossi*

Abstract

The aim of this work is to provide an analytical model to characterize the equilibrium points and the phase space associated with the singly‐averaged dynamics caused by the planetary oblateness coupled with the solar radiation pressure perturbations. A two‐dimensional differential system is derived by considering the classical theory, supported by the existence of an integral of motion comprising semi‐major axis, eccentricity and inclination. Under the single resonance hypothesis, the analytical expressions for the equilibrium points in the eccentricity‐resonant angle space are provided, together with the corresponding linear stability. The Hamiltonian formulation is also given. In the talk, a simple tool to visualize the structure of the phase space will be presented and the results obtained by considering the Earth as major oblate body will be showed. Finally, some considerations on the possible use of the proposed model for deorbiting strategies will be drawn.

This work was funded by the European Commission Horizon 2020, Frame‐ work Programme for Research and Innovation (2014‐2020), under the ReD‐ SHIFT project (grant agreement n. 687500) and under the project COMPASS (grant agreement n. 679086).

* IFAC‐CNR, Via Madonna del Piano 10, I‐50019 Sesto Fiorentino (FI), Italy † Associate professor, Department of Aerospace Science and Technology, Politecnico di Milano, Via la Masa 34, 20156 Milan, Italy


1st‐order secular resonances and L‐K mechanism for lunar perturbations in SSTO orbits

Denis Hautesserres*

Abstract

In 2016 we did a parametric study of the orbital lifetime of SSTO orbits where we produced first maps [5]. Our intuition has led us to know what are the mechanisms in action on these orbits. Now, we write the 1st‐order resonances network for lunar perturbations as Cook did [1] but for higher values of the Cook variable. And we put the SSTO orbits into the network to show how the orbits will behave. We begin to understand the life time maps of SSTOs, in particular the regions where the L‐K mechanism [2] [3] acts with or without the secular resonances. Then we learn that long lifetimes are due to the early and dense L‐K mechanism where the perigee suffer librations in the region empty of lunar secular resonances. We exhibit two precise lifetimes maps, thanks to the s/w HEOSAT[4]. Because now everybody knows, understands and can compute the future of SSTOs, it is time for companies to manage upper stages of their rockets with the aim to mitigate the number of debris in these regions of space with reducing a priori lifetimes. We can help them to do it.

Figure 2. SSTO orbits have apogees close to 28% pf the Earth‐Moon distance

References

[1] G. E. Cook, Luni‐Solar Perturbations of the Orbit of an Earth Satellite. The Geophysical Journal of the Royal Astronomical Society, Vol. 6, 271291 April (1962).

[2] M. L. Lidov, The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies. Planetary and Space Science 9, 719759 October (1962).

[3] Y. Kozai, Secular Perturbations of Asteroids with High Inclination and Eccentricity. The Astronomical Journal 67, 591‐598 November (1962).

[4] M. Lara, J. F. San‐Juan, D. Hautesserres, A semi‐analytical orbit propagator program for Highly Elliptical Orbits. In Proceedings of the 6th International Conference on Astrodynamics Tools and Techniques, ICATT. ESA, (2016).

[5] D. Hautesserres, J. F. San‐Juan, M. Lara, a parametric study of the orbital lifetime of super GTO and SSTO orbits based on semi‐analytical integration. Astrophysics and Space Science Proceedings 52, Stardust Final Conference, (2018).

* CNES, Toulouse, France, e‐mail: [email protected]


Diving into the 2g + h resonance

Jerome Daquin*, Ioannis Gkolias†, and Christos Efthymiopoulos‡

Abstract

The existence and interplay of multiple lunisolar secular resonances strongly affects the long term behavior of Medium Earth Orbits (MEOs) (Daquin et al., 2016; Celletti, 2016; Gkolias et al., 2016). In this contribution we recall a first order averaged model (expressed in closed formulas) and perform an analysis of the resonant structure and long‐term dynamics near the

inclination‐dependent‐only 2g+h resonance. The model takes into account the Earth’s

oblateness (J2) and perturbations by external bodies (Moon and Sun). After reducing the dynamics to integrable models recovering Breiter’s results (Breiter, 2001), we describe how fundamental features of the dynamics (position and stability of equilibrium points, amplitude of the separatrices) evolve according to certain parameters of the problem. We then discuss the need to consider important modifications to the simple picture provided by the integrable models, due to the appearance of strong chaos in the real dynamics. In that respect, we numerically compute various homoclinic tangles (under a hierarchy of dynamical models) using a weighted modification of the classical Fast Lyapunov Indicator (Guzzo and Lega, 2014) and we propose an estimate of the amplitude of the resonance in terms of its chaotic extent.

Keywords: Lunisolar secular resonance, Hamiltonian chaos, Space Situational Awareness.

References

[1] Breiter, S. (2001). Lunisolar resonances revisited. Celestial Mechanics and Dynamical Astronomy, 81:81–91.

[2] Celletti, A; Gales, C. (2016). A study of the lunisolar secular resonance 2 ω+ Ω=0. Front. Astr. Space Sci. 3: 11. doi: 10.3389/fspas, page 2.

[3] Daquin, J., Rosengren, A., Alessi, E., Deleflie, F., Valsecchi, G., and Rossi, A. (2016). The dynamical structure of the meo region: long‐term stability, chaos, and transport. Celestial Mechanics and Dynamical Astronomy, 124(4):335–366.

[4] Gkolias, I., Daquin, J., Gachet, F., and Rosengren, A. J. (2016). From order to chaos in earth satellite orbits. The Astronomical Journal, 152(5):119.

[5] Guzzo, M. and Lega, E. (2014). Evolution of the tangent vectors and localization of the stable and unstable manifolds of hyperbolic orbits by fast lyapunov indicators. SIAM Journal on Applied Mathematics, 74(4):1058–1086.

* Department of Mathematics ‘Tullio Levi‐Civita’, University of Padova, Padua, Italy, [email protected] † Department of Aerospace Science and Technology, Politecnico di Milano, Milan, Italy ‡ Research Center for Astronomy and Applied Mathematics, Academy of Athens


The integrated autocorrelation function as a dynamical chaos indicator

Matthieu Duroyon**, Jérôme Daquin† and Jérôme Perez*

Abstract

One aspect of modern application of chaos indicators is to reveal the dynamical structure of the phase space by portraying the distribution of ordered and chaotic motions. The so‐called ‘stability maps’ are obtained by computing the values of finite time chaos indicators on fine meshes of initial conditions of a specific domain of interest. Well known indicators include the Lyapunov Indicator, the Mean Exponential Growth factor of Nearby Orbits or the Fast Lyapunov Indicator. In this contribution, we show that a suited renormalisation of the Integrated Autocorrelation Function, IACF, (see Wytrzyszczak et al. (2007); Barrio et al. (2009)) is necessary to use the indicator in the context of stability maps. Contrarily to a large class of chaos indicators based on variational equations, the renormalised IACF, based on the correlation decay, doesn’t require to follow tangent vectors. The method can therefore be directly applied to time‐series, mappings or flows. As an additional application of the tool, we estimate the correlation times (Sprott, 2003) that we systematically compare to Lyapunov times in a secular model relevant for Medium Earth Orbits near the 2g + h resonance.

Keywords: Hamiltonian chaos, Chaos indicator, Correlation decay.

References

[1] Barrio, R., Borczyk, W., and Breiter, S. (2009). Spurious structures in chaos indicators maps. Chaos, Solitons & Fractals, 40(4):1697–1714.

[2] Sprott, J. C. (2003). Chaos and time‐series analysis. Oxford University Press.

[3] Wytrzyszczak, I., Breiter, S., and Borczyk, W. (2007). Regular and chaotic motion of high altitude satellites. Advances in Space Research, 40(1):134–142.

* ENSTA ParisTech, Department of Applied Mathematics, Palaiseau, France † Department of Mathematics ‘Tullio Levi‐Civita’, University of Padova, Padua, Italy


Representations of the force function of two rigid bodies in canonical variables

Aleksandr Zlenko* and Irina Ryabikova*

Abstract

Two new expansions of the force function of two rigid celestial bodies of finite sizes and arbitrary forms in the canonical variables of Delaunay‐Andoyer with any degree of accuracy as a partial sums of convergent eight dimensional trigonometric series are obtained. The terms of this series consist of products of impulses, Stok’s constants and trigonometric sine and cosine, the arguments of which contain the linear combinations of Delaunay‐Andoyer angular variables. The first expansion of the force function takes into account the different number of harmonics of each of the two bodies. The second expansion makes it possible to take into account all terms with an accuracy up to a certain fixed order determined by reciprocal of the major semi‐axis of their orbit. These representations of the force function can be applied in the canonical equations of motion in problems of celestial mechanics and astrodynamics.

Keywords: force function, canonical variables of Delaunay and Andoyer, trigonometric series.

References

[1] Zlenko A (2004) Translational‐rotational motion of the resonant satellites, Inter.Academy of information technologies, Moscow (in Russian).

[2] Šidlichovský M (1979) The force function of two general bodies II, Bull. Astron. Inst. Checosl., 30, No. 3, pp. 152‐155.

[3] Šidlichovský M (1981) The elimination of short periodic perturbations in the problem of two finite bodies, Bull. Astron. Inst. Checosl., 32, No. 3, pp. 159‐167.

[4] Smart W (1953) Celestial mechanics, Longmans, Green and CO, London‐New York‐Toronto.

[5] Duboshin G (1975) Nebesnaja Mekhanika. Osnovnye zadachi i metody. Nauka, Moscow (in Russian).

[6] Andoyer H (1923) Cours de Méchanique Céleste, Gauthier‐Villars, Paris.

[7] Aksenov E, Chazov V (2011) The model of earth artificial satellite motion. Main problem, basic algorithms, State Astron. Inst. Sternberg, Moscow State University (in Russian).

* Departments of High Mathematics and Descriptive Geometry, Moscow Automobile and Road Construction State Technical University (MADI), Moscow, Russia.


Long‐term evolution of extrasolar systems via normal forms

Marco Sansottera*, Anne‐Sophie Libert*, Ugo Locatelli* and Antonio Giorgill i*

Abstract

One of the most remarkable properties of extrasolar planets is their possibly high orbital eccentricities, in contrast to the quasi‐circular planetary orbits of the Solar System. The classical Laplace‐Lagrange secular theory uses the circular approximation as a reference, thus its applicability to extrasolar systems can be doubtful.

We aim to show here that perturbation theory reveals very efficient for describing the long‐term evolution of extrasolar systems. More precisely, we study the long‐term evolution of two‐planet extrasolar systems by extending the Laplace‐Lagrange theory. We identify three categories of systems: (i) secular systems, whose long‐term evolution is accurately described by an extension of the classical Laplace‐Lagrange theory to a high order in eccentricities; (ii) systems that are near a mean‐motion resonance, for which an extension of the Laplace‐Lagrange secular theory to order two in the masses is required; (iii) systems that are really close to or in a mean‐motion resonance, for which a resonant model has to be used.

In the first two cases, we determine the fundamental frequencies of the motion and compute precisely the long‐term evolution of the Keplerian elements with a totally analytical method, based on Lie series. Coming to the resonant systems, we show how the long‐term evolution can be accurately reproduced by including appropriate resonant combinations of the fast angles via a resonant normal form. This result extends the Laplace‐Lagrange secular approximation to resonant systems.

* University of Milan. Italy.

Scientific Computing Group (GRUCACI) University of La ... · The continuous development and upgrade of orbit ... Application of SAPO, the GMV’s Semi-Analytical Orbit Propagator,

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