Nowadays there is international consensus that space activities must be managed to minimize debris generation and risk. The paper presents a method for the end-of-life (EoL) disposal of spacecraft in high elliptical orbits (HEO). The time evolution of HEO is strongly affected by Earth's oblateness and luni-solar perturbation, and this can cause in the long-term to extended interferences with low Earth orbit (LEO) protected region and uncontrolled Earth re-entry. An EoL disposal concept that exploits the effect of orbital perturbations to reduce the disposal cost is presented. The problem is formulated as a multiobjective optimization problem, which is solved with an evolutionary algorithm. To explore at the best the search space a semi-analytical orbit propagator, which allows the propagation of the orbit motion for 100 years in few seconds, is adopted. The EoL disposal of the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL) mission is used as a practical test-case to show the effectiveness of the proposed methodology.
The possibility of having collision between a satellite and a space debris or another satellite is becoming frequent. The amount of propellant is directly related to a satellite's operational lifetime and revenue. Thus, collision avoidance maneuvers should be performed in the most efficient and effective manner possible. In this work the problem is formulated as a multi-objective optimization. The first objective is the the?v, whereas the second and third one are the collision probability and relative distance between the satellite and the threatening object in a given time window after the maneuver. This is to take into account that multiple conjunctions might occur in the short-term. This is particularly true for the GEO regime, where close conjunction between a pair of object can occur approximately every 12h for a few days. Thus, a CAM can in principle reduce the collision probability for one event, but significantly increase it for others. Another objective function is then added to manage mission constraint. To evaluate the objective function, the TLE are propagated with SGP4/SDP4 to the current time of the maneuver, then the?v is applied. This allow to compute the corresponding "modified" TLE after the maneuver and identify (in a given time window after the CAM) all the relative minima of the squared distance between the spacecraft and the approaching object, by solving a global optimization problem rigorously by means of the verified global optimizer COSY-GO. Finally the collision probability for the sieved encounters can be computed. A Multi-Objective Particle Swarm Optimizer is used to compute the set of Pareto optimal solutions. The method has been applied to two test cases, one that considers a conjunction in GEO and another in LEO. Results show that, in particular for the GEO case, considering all the possible conjunctions after one week of the execution of a CAM can prevent the occurrence of new close encounters in the short-term.
Carta R, Armellin R, Di Lizia P, Bernelli-Zazzera F, Magnani P, Re E (2014) DEM simulation of sampling tool mechanisms for low gravity bodies, Proceedings of the 64th International Astronautical Congress 2 pp. 1276-1281
In future exploration mission to low gravity bodies (e.g. a Mars moon or a near-Earth asteroid) it is planned to collect more than 100 grams of regolith (dust plus cm-sized pebbles) and return them to Earth for further ground-based analysis. In past Near-Earth Asteroid and Marco Polo and current Marco Polo-R studies several sampling tools have been proposed but there is no single technology for low-gravity sampling that has undergone a rigorous engineering assessment, aiming at proving the ability of the sampler to collect material in any envisaged situation. This is the purpose of this activity. The use of Discrete Element Method (DEM) to investigate mechanical properties of geomaterials is growing fast and their applications in geotechnics have become almost systematic. DEM granular approach suites the problem of studying the soil dynamics during the soil-tool interaction; moreover the investigation of the tool effectiveness in sampling under the influence of desired parameters (tool geometry, tool motion, soil properties) can be addressed. On the other hand, the DEM model of the soil needs a lot of effort to set up the particles interaction in order to faithfully represent the real work environment. The DEM is implemented to realize an affordable and reliable tool useful to investigate the sampling device dynamics in soil sampling activities to support the sampling tool concepts identification, trade off and selection. Comparing real experiment data to numerical models proves the quantitative capabilities of the DEM tool. The implemented tool addresses both the effectiveness evaluation of the sampling device concepts and the main solicitations estimation. This study addresses the following main steps: review of requirements and soil parameters identification, soil specimen modelling, sampling tool concept modelling, dynamic simulation of soil sampling, sensitivity analysis of dynamic simulation to soil parameters and environment variables. Copyright © 2013 by the International Astronautical Federation. All rights reserved.
This paper introduces and combines two novel techniques. Firstly, we introduce an efficient numerical method for the propagation
of entire sets of initial conditions in the phase space and their associated phase space densities based on Differential
Algebra (DA) techniques. Secondly, this DA density propagator is applied to a DA-enabled implementation of Semi-Analytical
(SA) averaged dynamics, combining for the first time the power of the SA and DA techniques.
While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system
under consideration, the particular combination of DA techniques with SA equations yields a fast and accurate method to
propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time.
This enables the study of the long-term behavior of particles subjected to the given dynamics.
To demonstrate the effectiveness of the proposed method, the evolution of a cloud of high area-to-mass objects in Medium
Earth Orbit is reproduced considering the effects of solar radiation pressure, the Earth?s oblateness and luni-solar perturbations.
The computational efficiency is demonstrated by propagating 10; 000 random samples taking snapshots of their state
and density at evenly spaced intervals throughout the integration. The total time required for a propagation for 16 years in the
dynamics is on the order of tens of seconds on a common desktop PC.
Wittig A, Colombo C, Armellin R (2015) Density of high area-to-mass objects in geostationary and medium Earth orbits through semi-analytical equations and differential algebra, Proceedings of the 65th International Astronautical Congress 2014 (IAC 2014) 7 pp. 4599-4609 International Astronautical Federation (IAF)
This paper introduces and combines two novel techniques. Firstly, we introduce an efficient numerical method for the propagation of entire sets of initial conditions in the phase space and their associated phase space densities based on Differential Algebra (DA) techniques. Secondly, this DA density propagator is applied to a DA-enabled implementation of Semi-Analytical (SA) averaged dynamics, combining for the first time the power of the SA and DA techniques. While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with SA equations yields a fast and accurate method to propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics. To demonstrate the effectiveness of the proposed method, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the effects of solar radiation pressure, the Earth's oblateness and luni-solar perturbations. The computational efficiency is demonstrated by propagating 10.000 random samples taking snapshots of their state and density at evenly spaced intervals throughout the integration. The total time required for a propagation for 16 years in the dynamics is on the order of tens of seconds on a common desktop PC.
Armellin R, Di Lizia P, Bernelli Zazzera F, Berz M (2010) Nonlinear mapping of uncertainties: a differential algebraic approach,
A method for the nonlinear propagation of uncertainties
in celestial mechanics based on differential
algebra is presented. The arbitrary order
Taylor expansion of the flow of ordinary differential
equations with respect to the initial condition
delivered by differential algebra is exploited to
implement an accurate and computationally efficient
Monte Carlo algorithm, in which thousands
of pointwise integrations are substituted
by polynomial evaluations. The algorithm is applied
to study the close encounter of asteroid
Apophis with our planet in 2029. To this aim,
we first compute the high order Taylor expansion
of Apophis? close encounter distance from
the Earth by means of map inversion and composition;
then we run the proposed Monte Carlo
algorithm to perform the statistical analysis.
Armellin R, Dowlat D, Lavagna M (2011) Lunar soft landing trajectory optimization in a 6DoF dynamical model, Conference of the European Aerospace Socities (CEAS) 2011 Post-Conference Proceedings pp. 1211-1220 Confine Edizioni
An algorithm for the optimization of a lunar soft-landing trajectory is presented. A 6DOF modeling of the dynamics is adopted together with an accurate description of the Moon gravity field. The problem is faced as a direct optimization problem with the goal of obtaining a vertical landing whilst minimizing the overall fuel consumption. The descent trajectory is supposed to start from the periselenium of a low Moon orbit. Four optimization phases are considered. Each phase is characterized by a different set of optimization variables, constraints, and increasing level of complexity. In the first phase the thrust direction is optimized considering the translational motion of the lander only. Furthermore, no throttle capability is considered. In the second phase the thrust direction is fixed in the spacecraft body reference frame. The proper thrust orientation is obtained by optimizing the control torques supplied to the lander by the attitude sub-system. In the third phase the thrust magnitude is optimized too, and the constraint of landing on specific site is added. Furthermore, more restrictive constraints on the final velocities (linear and angular) are set. Finally, in the fourth phase a more accurate gravitational model of Moon that includes the main harmonics is considered. The algorithm is tested on two different landing scenarios. One describes a landing near the north pole area for a mission whose goal is to visit the craters where recently the presence of water has been discovered. The second one considers a landing in an area close to the Moon's equator, and it is inspired by Google Lunar X-prize.
Padula MP, Armellin R, Di Lizia P, Lavagna M (2014) A simulation tool for space situational awareness: Near Earth Objects, Proceedings of the 64th International Astronautical Congress 7 pp. 5592-5605
A simulation tool for Near Earth Objects (NEO) orbit determination by means of optical measurements is presented. The peculiarity of the simulator is the use of high order methods based on Differential Algebra (DA) techniques. State-of-the art tools are mostly based on either linear methods or nonlinear Monte Carlo simulations. The main advantage of linear methods stays in their problem simplification, but their accuracy drops off rapidly for large uncertainty sets. Classical Monte Carlo simulations provide true statistics, but are computationally intensive. DA techniques are used to overcome the above issues, supplying the tools to compute arbitrary order derivatives of functions within a computer environment with limited computational effort. The availability of the high order Taylor expansions is exploited to manage problem uncertainties. The tool includes a simulator of optical observations, DA-based algorithms for NEO preliminary and accurate orbit determination. Angular measurements simulation is based on the propagation in time of initial asteroid state through multi-body dynamics; aberration, precession and nutation effects may be taken into account by the simulator. Preliminary orbit determination (POD) is based on Lambert's and Kepler's problems and uses the real solutions of the classical Gauss method 8th order polynomial as first guesses of an iterative procedure. A better convergence with respect to Gauss method is achieved. Observations uncertainties are analytically mapped to the phase space as high-order multivariate Taylor polynomials. When more than three observations are available, the tool applies a high order Extended Kalman filter, initialized by the POD solution. The uncertainties related to the POD are propagated forward in time by exploiting the high order expansion of the flow of the dynamics. Thus, the initial covariance is nonlinearly and analytically propagated up to the next measurement. The performance of the tool is analysed by running the algorithms on a list of real Near Earth Asteroids and simulated topocentric observations.
Optimal feedback control is classically based on linear approximations, whose accuracy drops off rapidly in highly nonlinear dynamics. Several nonlinear optimal feedback control strategies have appeared in recent years. Among them, differential algebraic techniques have already been used to tackle nonlinearities by expanding the solution of the optimal control problem about a reference trajectory and reducing the computation of optimal feedback control laws to the evaluation of high order polynomials. However, the resulting high order method could not handle control saturation constraints, which remain a critical facet of nonlinear optimal feedback control. This work introduces the management of saturating actuators in the differential algebraic method. More specifically, the constraints are included in the optimal control problem formulation and differential algebra is used to expand the associated optimal bang-bang solution with respect to initial and terminal conditions. Optimal feedback control laws for thrust direction and switching times are again computed by evaluating the resulting polynomials. Illustrative applications are presented in the frame of the optimal low-thrust transfer to asteroid 1996 FG3.
Di Lizia P, Massari M, Losacco M, Bianchi G, Mattana A, Pupillo G, Bortolotti C, Roma M, Morselli A, Armellin R, Pupillo G, Magro A, Cutajar D, Portelli C, Reali M (2017) Performance assessment of the multibeam radar sensor birales for space surveillance and tracking, Proceedings of 7th European Conference on Space Debris
Near-Earth space has become progressively more crowded in active satellites, inactive spacecraft and debris. Consequently, an international effort is currently being devoted to improving the performance of the network of optical and radar sensors for space objects monitoring. Within this framework, the use of the novel bistatic radar sensor BIRALES is investigated in this work, which makes use of a multibeam receiver. The tailored orbit determination algorithm is described, which receives as input the data processed by the acquisition system, that digitally assembles measured radar echoes. The performances of the orbit determination process are assessed on a set of numerical simulations carried out on the NORAD catalogue, using a dedicated simulator of the sensor.
A sixth-order accurate scheme is presented for the solution of ODE systems supplemented by two-point boundary conditions. The proposed integration scheme is a linear multi-point method of sixth-order accuracy successfully used in fluid dynamics and implemented for the first time in astrodynamics applications. A discretization molecule made up of just four grid points attains a O(h 6) accuracy which is beyond the first Dahlquist's stability barrier. Astrodynamics applications concern the computation of libration point halo orbits, in the restricted three- and four-body models, and the design of an optimal control strategy for a low thrust libration point mission.
Morselli A, Armellin R, Di Lizia P, Zazzera FB, Salerno E, Bianchi G, Montebugnoli S, Magro A, Adami KZ (2015) Orbit determination of space debris using a Bi-static radar configuration with a multiple-beam receiver, Proceedings of the 65th International Astronautical Congress 2014 (IAC 2014). 3 pp. 1774-1784
Curran Associates, Inc.
In this work the use of a multi-beaming radar system is analyzed and a possible setup of a closed loop system (i.e. from measurement and data acquisition to orbit determination) is described. The Orbit Determination (OD) algorithms are specialized for a bistatic radar configuration where the Medicina Northern Cross radio-telescope (owned by the University of Bologna-Italy) is considered as a receiver. The Northern Cross is composed of two perpendicular arms: the E/W arm is 564 m long and consists in a single cylindrical antenna with a width of 29.4 m, whereas the N/S arm is made of 64 parallel antennas with a length of 22.6 m and a width of 7.5 m. The collecting area reaches 27,400 sqm and, by considering a complete upgrade of the radar with the installation of new receivers on the focal lines, up to 22,880 possible theoretical independent beams could cover the field-of-view of 55.47 (E/W) deg x 1.8 (N/S) deg. By looking at the sequence of beams that are illuminated, it is thus possible to estimate, with an higher level of detail with respect to the single-beam system, the ground track of the transiting object. Given this peculiar system, tailored orbit determination algorithms have to be developed. The orbit determination algorithm receives as input the data processed by the acquisition system, that digitally assembles measured radar echoes, using Fast Fourier Transform, to provide the signal for each beam. These inputs are the measured Doppler shift, time delay, the illumination time and measured power intensity associated to each beam. By combining these information with the knowledge of beam distribution and pointing it is possible to refine the orbital parameters of known objects or to perform a preliminary OD. A few LEO objects are considered to generate simulated data that are then used to feed the developed OD algorithms. In this way the performances of the algorithms can be tested and the effectiveness of this innovative configuration for space debris measurements, that couples a bistatic radar and a multi-beaming receiver, can be assessed.
The aerogravity-assist maneuver is proposed as a tool to improve the efficiency of the gravity assist, because due to the interaction with the planetary atmosphere, the angular deviation of the velocity vector can be definitely increased. Even though the drag reduces the spacecraft velocity, the overall Av gain could be substantial for a high-lift-to-drag vehicle. A previous study addressed the three-dimensional dynamic modeling and optimization of the maneuver, including heliocentric plane change, heating rate, and structural load analysis. A multidisciplinary study of aerogravity assist is proposed, focusing on coupled trajectory and vehicle shape optimization. A planar aerogravity assist of Mars is selected as a test case, with the aim of maximizing the vehicle heliocentric velocity and limiting the heating rate experienced during the atmospheric pass. A multiobjective approach is adopted, and a particle swarm optimization algorithm is chosen to detect the set of Pareto-optimal solutions. The study includes a further refinement of the trajectory for three significant shapes belonging to the Pareto curve. The associated optimal control problem is solved by selecting a direct-method approach. The dynamics are transcribed into a set of nonlinear constraints, and the arising nonlinear programming problem is solved through a sequential quadratic programming solver. Copyright © 2007 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
Rasotto M, Morselli A, Wittig A, Massari M, Di Lizia P, Armellin R, Valles C, Ortega G (2016) Differential algebra space toolbox for nonlinear uncertainty propagation in space dynamics,
This paper is aimed at presenting a new tool developed by Dinamica, with the support of ESA, for the efficient non-linear propagation of uncertainties in space dynamics. The newly implemented software is based on Differential Algebra, which provides a method to easily extend the existing linearization techniques and allows the implementation of efficient arbitrary order methods. These theoretical concepts represent the building blocks over which the Differential Algebra Space Toolbox is implemented. The application areas for the tool are plenty. To illustrate the power of the method in general and to give the user a better understanding of the various features, several different examples in the field of astrodynamics and space engineering are presented.
Lidtke AA, Lewis HG, Armellin R (2015) Impact of high-risk conjunctions on Active Debris Removal target selection, Advances in Space Research 56 (8) pp. 1752-1764
All rights reserved.Space debris simulations show that if current space launches continue unchanged, spacecraft operations might become difficult in the congested space environment. It has been suggested that Active Debris Removal (ADR) might be necessary in order to prevent such a situation. Selection of objects to be targeted by ADR is considered important because removal of non-relevant objects will unnecessarily increase the cost of ADR. One of the factors to be used in this ADR target selection is the collision probability accumulated by every object. This paper shows the impact of high-probability conjunctions on the collision probability accumulated by individual objects as well as the probability of any collision occurring in orbit. Such conjunctions cannot be predicted far in advance and, consequently, not all the objects that will be involved in such dangerous conjunctions can be removed through ADR. Therefore, a debris remediation method that would address such events at short notice, and thus help prevent likely collisions, is suggested.
In this chapter, a method to assess the occurrence of impacts between
objects (either spacecraft or space debris) orbiting around the Earth is presented.
The method is based on the computation of the minimum distance between two
evolving orbits by means of a rigorous global optimizer. Analytical solutions of
artificial satellite motion are utilized to account for perturbative effects of Earth?s
zonal harmonics, atmospheric drag, and third body. It is shown that the method can
effectively compute the intersection between perturbed orbits and hence identify
pairs of space objects on potentially colliding orbits. Test cases considering sunsynchronous,
low perigee and earth-synchronous orbits are presented to assess the
performances of the method.
An effective method for the design of fuel-optimal transfers in two- and three-body dynamics is presented. The optimal control problem is formulated using calculus of variation and primer vector theory. This leads to a multi-point boundary value problem (MPBVP), characterized by complex inner constraints and a discontinuous thrust profile. The first issue is addressed by embedding the MPBVP in a parametric optimization problem, thus allowing a simplification of the set of transversality constraints. The second problem is solved by representing the discontinuous control function by a smooth function depending on a continuation parameter. The resulting trajectory optimization method can deal with different intermediate conditions, and no a priori knowledge of the control structure is required. Test cases in both the two- and three-body dynamics show the capability of the method in solving complex trajectory design problems.
Di Mauro G, Di Lizia P, Armellin R, Lavagna M (2012) Nonlinear control of leader-follower formation flying, Advances in the Astronautical Sciences, vol. 145: Proceedings of the 1st IAA Conference on Dynamics and Control of Space Systems (DyCoSS) 145 pp. 215-230
This paper considers the problem of relative motion control involved in a leader-follower formation keeping mission. More specifically, center of mass dynamics of two Earth orbiting satellite is modeled, including the nonlinearity due to Earth oblateness. Next, the differential algebra is exploited to compute an high order Taylor expansion of the State-Dependent Riccati Equation (SDRE) solution. This new approach reduces the computational cost of the online Algebraic Riccati Equation solution required by SDRE algorithm; in fact, the differential algebraic formulation gives a polynomial representation which can be directly evaluated for SDRE solutions or exploited to define an initial first guess for iterative SDRE algorithms.
Armellin R, De Lizia P, Lavagna M (2011) Preliminary orbit determination: uncertainty management via Taylor expansions, Conference of the European Aerospace Socities (CEAS) 2011 Post-Conference Proceedings
Lunghi P, Lavagna M, Armellin R (2015) A semi-analytical guidance algorithm for autonomous landing, Advances in Space Research 55 (11) pp. 2719-2738 Elsevier
One of the main challenges posed by the next space systems generation is the high level of autonomy they will require. Hazard Detection and Avoidance is a key technology in this context. An adaptive guidance algorithm for landing that updates the trajectory to the surface by means of an optimal control problem solving is here presented. A semi-analytical approach is proposed. The trajectory is expressed in a polynomial form of minimum order to satisfy a set of boundary constraints derived from initial and final states and attitude requirements. By imposing boundary conditions, a fully determined guidance profile is obtained, function of a restricted set of parameters. The guidance computation is reduced to the determination of these parameters in order to satisfy path constraints and other additional constraints not implicitly satisfied by the polynomial formulation. The algorithm is applied to two different scenarios, a lunar landing and an asteroidal landing, to highlight its general validity. An extensive Monte Carlo test campaign is conducted to verify the versatility of the algorithm in realistic cases, by the introduction of attitude control systems, thrust modulation, and navigation errors. The proposed approach proved to be flexible and accurate, granting a precision of a few meters at touchdown.
A method to deal with uncertainties in initial orbit determination (IOD) is presented. This is based on the use of Taylor differential algebra (DA) to nonlinearly map uncertainties from the observation space to the state space. When a minimum set of observations is available, DA is used to expand the solution of the IOD problem in Taylor series with respect to measurement errors. When more observations are available, high order inversion tools are exploited to obtain full state pseudo-observations at a common epoch. The mean and covariance of these pseudo-observations are nonlinearly computed by evaluating the expectation of high order Taylor polynomials. Finally, a linear scheme is employed to update the current knowledge of the orbit. Angles-only observations are considered and simplified Keplerian dynamics adopted to ease the explanation. Three test cases of orbit determination of artificial satellites in different orbital regimes are presented to discuss the feature and performances of the proposed methodology.
An adaptive guidance algorithm for close approach to and precision landing on uncooperative low-gravity objects (e.g. asteroids) is proposed. The trajectory, updated by means of a minimum fuel optimal control problem solving, is expressed in a polynomial form of minimum order to satisfy a set of boundary constraints from initial and final states and attitude requirements. Optimal guidance computation, achieved with a simple two-stage compass search, is reduced to the determination of three parameters, time-of-flight, initial thrust magnitude and initial thrust angle, according to additional constraints due to actual spacecraft architecture. A NEA landing mission case is analyzed.
The paper presents a new semi-analytical technique for the propagation of near-Earth satellite motion. The approach uses differential algebra techniques to compute the high order expansion of the solution of the system?s ordinary differential equation for one orbital revolution, referred to as the transfer map. Once computed, a single high order transfer map (HOTM) can be reused to map an initial condition, or a set of initial conditions, forward in time for many revolutions. The only limiting factor is that the mapped objects must stay close to the reference orbit such that they remain within the region of validity of the HOTM. The performance of the method is assessed through a set of test cases in which both autonomous and non-autonomous perturbations are considered, including the case of continuously propelled trajectories.
Morselli A, Di Lizia P, Bianchi G, Bortolotti C, Montebugnoli S, Naldi G, Perini F, Pupillo G, Roma M, Schiaffino M, Mattana A, Salerno E, Magro A, Adami KZ, Armellin R, Sergiusti AL, Villadei W, Dolce F, Reali M, Paoli J (2015) A new high sensitivity radar sensor for space debris detection and accurate orbit determination, 2nd IEEE International Workshop on Metrology for Aerospace, MetroAeroSpace 2015 - Proceedings pp. 562-567 IEEE
As the amount of space debris orbiting the earth is continuously increasing, it is becoming essential to monitor and predict the debris' trajectories in order to avoid collision that could threaten space missions, i.e. operative satellites or manned spacecraft. An innovative bistatic radar system, that couples multi-beaming and ranging techniques, is proposed in this work as an instrument for the observation of Earth-orbiting objects. The main characteristics of the sensor are described. In addition, tailored algorithms have been implemented to perform orbit determination using the peculiar outputs provided by the sensor. The results of numerical simulations are illustrated to assess its performances for space debris detection and orbit determination.
Di Lizia P, Armellin R, Bernelli Zazzera F, Berz M (2008) High order expansion of the solution of two-point boundary value problems using differential algebra: applications to spacecraft dynamics, Fields Institute Communications American Mathematical Society
Two-point boundary value problems appear frequently in
space trajectory design. A remarkable example is represented by the
Lambert?s problem, where the conic arc linking two fixed positions in
space in a given time is to be characterized in the frame of the two-
body problem. However, a certain level of approximation always affects
the dynamical models adopted to design the nominal trajectory of a
spacecraft. Dynamical perturbations usually act on the spacecraft in
real scenarios, deviating it from the desired nominal trajectory. Conse-
quently, the boundary conditions assumed for the nominal solutions are
usually affected by uncertainties and errors. Suitable techniques must
be developed to quickly compute correction maneuvers to compensate
for such errors in practical applications. This work proposes differential
algebra as a valuable tool to face the previous problem. An algorithm is
presented, which is able to deliver the arbitrary order Taylor expansion
of the solution of a two-point boundary value problem about an available
nominal solution. The mere evaluation of the resulting polynomials en-
ables the design of the desired correction maneuvers. The performances
of the algorithm are assessed by addressing typical applications in the
field of spacecraft dynamics, such as the simple Lambert?s problem and
the station keeping of a spacecraft around a nominal halo orbit.
Armellin R, Dilizia P, Crepaldi M, Bernelli-Zazzera F, Ercoli Finzi A (2014) Scientific use of the sampler, drill and distribution subsystem (SD2), JBIS - Journal of the British Interplanetary Society 67 (11-12) pp. 426-433
Rosetta is the third cornerstone mission of the European Space Agency scientific program "Horizon 2000". Rosetta will be the first spacecraft to orbit around a comet nucleus. It was launched in March 2004 and will reach the comet 67P/Churymov- Gerasimenko in 2014. A lander (Philae) will be released and land on the comet surface for in-situ investigation. One of the key subsystems of the lander Philae is the Sampler, Drill and Distribution (SD2) subsystem. SD2 provides in-situ operations devoted to soil drilling, samples collection, and their distribution to two evolved gas analyzers (COSAC and PTOLEMY) and one imaging instrument (ÇIVA). Recent studies have proven the existence of a correlation between the drill behavior during perforation and the mechanical characteristics of the cometary soil. This outlines the possibility of using SD2 not only as a tool to support other instruments, but also as a scientific instrument itself. In this paper the possibility of using the drill as a quasi-static penetrator is presented. Within this approach, laboratory tests on glass-foam specimens of different porosity show that the drill behaviour during penetration can be exploited for cometary soil characterization.
Lavagna M, Parigini C, Armellin R (2006) Pso algorithm for planetary atmosphere entry vehicles multidisciplinary guidance design, Collection of Technical Papers - AIAA/AAS Astrodynamics Specialist Conference, 2012 1 pp. 172-187 American Institute of Aeronautics and Astronautics
The paper presents a possible approach to define - within the same global optimization the guidance history and the configuration of each different flight regime an atmospheric Entry-Descent-Landing (EDL) vehicle has to deal with. Precision landing constraints as well as inertial and thermal loads containment are considered. The optimization is focused on detecting a set of possible preliminary solutions to be further investigated by a local finer optimizer. A partially revisited Particle Swarm Optimization technique has been here successfully applied to deal with both a multiobjective and distributed optimization architecture specifically thought to cope with the complexity of the proposed problem. The proposed approach turned out to be powerful in identifying, quite rapidly, a set of feasible good solutions both from the guidance and the configuration point of view for each of the aerodynamics phases the probe goes through.
Armellin R, Valli M, Di Lizia P, Lavagna MR, Zanetti R (2013) Nonlinear filtering based on Taylor differential algebra, Advances in the Astronautical Sciences vol.148: Proceedings of the 23rd AAS/AIAA Space Flight Mechanics Meeting held February 10?14, 2013, Kauai, Hawaii 148 pp. 3-21 American Astronautical Society
The problem of nonlinear filtering represents a crucial issue in celestial mechanics. In this paper a high-order filter based on differential algebra is presented. The proposed filtering algorithm is based on nonlinear mapping of statistics and a linear update scheme, in which only the probability density function of the measurement errors are constrained to be Gaussian. No hypothesis on the state probability density function is made. The case of an Earth-orbiting spacecraft in a two-body problem frame is considered as an example to demonstrate the general feature and performance of the filter. A comparison with the extended and unscented Kalman filters is also included.
The paper presents the Particle Swarm Optimization (PSO) technique as a possible approach to identify the globally optimal guidance history and configuration deployment sequence within different flight regimes an atmospheric entry-descent-landing (EDL) vehicle with a variable architecture has to deal with. The aerodynamics data set is dynamically generated according to the configuration solution identified by the current optimization iteration. The flight history is split according to the flight regime experienced by the vehicle in parallel Particle Swarm Optimization architectures to better exploit and visit the wide solution space. A 3D dynamics is modelled to generate the differential constraints the optimization must answer to. The optimization criteria vector is focused on multidisciplinary aspects such as the heat load experienced together with the precision landing conditions. Simulations on Mars atmosphere entry revealed the efficiency of the optimization architecture to detect the related Pareto front.
The manuscript addresses the problem of computing conjunctions either between Near Earth Objects and our planet or space debris and operative spacecraft. The problem is formulated as a global optimization problem, solved rigorously using Taylor models. With this technique, narrow bounds of the objective function are computed on sub-portions of the search space by combining high order Taylor approximations of the function with interval enclosures of the remainder terms. An algorithm based on differential algebra is then presented to nonlinearly describe the effect that uncertainties on the initial states produce on the time and distance of closest approach. This is represented with high order Taylor maps which can be potentially used for the implementation of innovative algorithms for risk assessment, avoiding the main assumptions of current approaches. Asteroid Apophis and a threatening condition between two geostationary satellites are considered as examples to analyze the features of the proposed methods.
Ferraro D, Armellin R, Di Lizia P, Lavagna M (2008) Sensitivity analysis and guidance for EDL phases by means of differential algebra, Proceedings of the 59th International Astronautical Congress 2008 (IAC 2008) Curran Associates, Inc.
Armellin R, Lavagna M, Starkey RP, Lewis MJ (2006) Aero-gravity assist maneuvers: Coupled trajectory and vehicle shape optimization, Collection of Technical Papers - AIAA/AAS Astrodynamics Specialist Conference, 2006 1 pp. 255-269 American Institute of Aeronautics and Astronautics
The aero-gravity assist maneuver is proposed as a tool to improve the efficiency of the gravity assist as, due to the interaction with the planetary atmosphere, the angular deviation of the velocity vector can be definitely increased. Even though the drag reduces the spacecraft velocity, the overall v gain could be remarkable whenever a high lift-to-drag vehicle flies. A previous study addressed the 3D dynamic modeling and optimization of the maneuver including heliocentric plane change, heating rate, and structural load analysis. A multidisciplinary study of aero-gravity assist is proposed, focusing on coupled trajectory and vehicle shape optimization. A planar aero-gravity assist of Mars has been selected as a test case with the aim of maximizing the vehicle heliocentric velocity and limiting the heating rate experienced during the atmospheric path. A multiobjective approach has been adopted, and a particle swarm optimization algorithm has been chosen to detect the set of Pareto optimal solutions. The study includes a further refinement of the trajectory for three significant shapes belonging to the Pareto curve. The associated optimal control problem has been solved by selecting a direct method approach. The dynamics has been transcribed into a set of nonlinear constraints and the arising non linear programming problem has been solved through a sequential quadratic programming solver.
Gondelach D, Lidtke A, Armellin R, Colombo C, Lewis H, Funke Q, Flohrer T (2016) Re-entry prediction of spent rocket bodies in GTO, Spaceflight Mechanics volume 158: Proceedings of the 26th AAS/AIAA Space Flight Mechanics Meeting held February 14?18, 2016, Napa, California, U.S.A. 158 pp. 2635-2654
American Astronautical Society (AAS)
Spent upper stages are bodies consisting of components likely to survive re-entry, for example propellant tanks. Therefore, the re-entry of upper stages might be associated with high on-ground casualty risk. This paper presents a tool for re-entry prediction of spent rocket bodies in GTO based exclusively on Two Line Element set (TLE) data. TLE analysis and filtering, spacecraft parameters estimation, and combined state and parameters estimation are the main building blocks of the tool. The performance of the tool is assessed by computing the accuracy of the re-entry prediction of 92 GTO objects, which re-entered in the past 50 years
Armellin R, Lavagna M, Ercoli Finzi A (2005) Aero-gravity assist maneuvers for interplanetary missions: controlled dynamics modelling and optimization,
Rasotto M, Armellin R, Di Lizia P, Bernelli-Zazzera F (2014) Optimal low-thrust transfers in two-body and three-body dynamics, Proceedings of the 64th International Astronautical Congress 7 pp. 5230-5244
This paper describes an efficient and robust optimal control algorithm for the design of fuel-optimal low-thrust interplanetary transfers. The solution is obtained with an indirect optimization approach, which has been selected to minimize the number of unknowns and to limit the computational effort. The optimization algorithm can deal with different intermediate conditions, such as flybys, rendezvous, or multiple gravity assists. Moreover, within the formulation adopted, no a priori knowledge of the control structure is required. The application of calculus of variations leads to a Multi-Point Boundary Value Problem (MP-BVP), characterized by complex inner constraints, which has been solved by means of an indirect multiple shooting method. Some effective techniques to increase the robustness of the algorithm and to overcome numerical difficulties are introduced, followed by the presentation of some test cases to assess the overall performances of the software-tool.
Valli M, Armellin R, Di Lizia P, Lavagna MR (2012) Nonlinear filtering methods for spacecraft navigation based on differential algebra, Advances in the Astronautical Sciences, vol. 145: Proceedings of the 1st IAA Conference on Dynamics and Control of Space Systems (DyCoSS) 145 pp. 259-274 Univelt Inc.
The nonlinear filtering problem plays an important role in various space-related applications and especially in orbit determination and navigation problems. Differential algebraic (DA) techniques are here proposed as a valuable tool to implement the higher-order numerical and analytic extended Kalman filters. Working in the DA framework allows us to consistently reduce the required computational effort without losing accuracy. The performance of the proposed filters is assessed on different orbit determination problems with realistic orbit uncertainties. The case of nonlinear measurements is also considered. Numerical simulations show the good performance of the filter in case of both complex dynamics and highly nonlinear measurement problems.
An efficient method is proposed to evaluate the Vapour-Liquid Equilibrium (VLE) curve for complex multi-parameter technical and reference thermodynamic equations of state, in connection with Computational Fluid Dynamics (CFD) simulations of compressible flows of real gases. Differential algebra techniques are used to obtain an approximation of the VLE curve from the reference equation of state of carbon dioxide. Seven fourth-order Taylor polynomials are required to approximate the VLE curve for a reduced pressure between 0.7 and 1, with an error on density below 0.04%, except near the critical point where the error is around 0.1%. The proposed approach is proved to be a suitable alternative to standard Look-Up Table (LUT) techniques, with comparable accuracy and computational burden. Moreover, the explicit use of the model analytical expression in the determination of the polynomial expansions allows to reduce the number of expansion poles and it will possibly simplify the approximation of different fluids, including mixtures. © 2014 Elsevier B.V. All rights reserved.
Di Lizia P, Armellin R, Bernelli Zazzera F, Berz M (2011) High order optimal feedback control of lunar landing and rendezvous trajectories, Conference of the European Aerospace Socities (CEAS) 2011 Post-Conference Proceedings. pp. 1221-1230 Confine Edizioni
Optimal feedback control is classically based on
linear approximations, whose accuracy drops off
rapidly in highly nonlinear dynamics. A high
order optimal control strategy is proposed in this
work, based on the use of differential algebraic
techniques. In the frame of orbital mechanics,
differential algebra allows the dependency of the
spacecraft state on initial conditions and environmental parameters to be represented by high
order Taylor polynomials. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Based on the reduction of
the optimal control problem to an equivalent two-point boundary value problem, differential algebra is used in this work to compute the high order
expansion of the solution of the optimal control
problem about a reference trajectory. New optimal control laws for displaced initial states are
then obtained by the mere evaluation of polynomials.
A high order optimal control strategy is proposed in this work, based on the use of differential algebraic techniques. In the frame of orbital mechanics, differential algebra allows to represent, by high order Taylor polynomials, the dependency of the spacecraft state on initial conditions and environmental parameters. The resulting polynomials can be manipulated to obtain the high order expansion of the solution of two-point boundary value problems. Since the optimal control problem can be reduced to a two-point boundary value problem, differential algebra is used to compute the high order expansion of the solution of the optimal control problem about a reference trajectory. Whenever perturbations in the nominal conditions occur, new optimal control laws for perturbed initial and final states are obtained by the mere evaluation of polynomials. The performances of the method are assessed on lunar landing, rendezvous maneuvers, and a low-thrust Earth-Mars transfer.
Armellin R, Lavagna M (2007) Multidisciplinary optimization of aerocapture maneuvers,
The paper investigates the problem of nonlinear filtering applied to spacecraft navigation. Differential algebraic (DA) techniques are proposed as a valuable tool to implement the higher-order numerical and analytic extended Kalman filters. Working in the DA framework allows us to consistently reduce the required computational effort without losing accuracy. The performance of the proposed filters is assessed on different orbit determination problems with realistic orbit uncertainties. The case of nonlinear measurements is also considered. Numerical simulations show the good performance of the filter in case of both complex dynamics and highly nonlinear measurement problems.
Current approaches to uncertainty propagation in astrodynamics mainly refer to linearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails in highly nonlinear dynamics or long term propagation. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial of the current state is split in two polynomials when its accuracy reaches a given threshold. The resulting polynomials accurately track uncertainties, even in highly nonlinear dynamics. The method is tested on the propagation of (99942) Apophis post-encounter motion.
Principe G, Armellin R, Lewis H (2016) Confidence region of least squares solution for single-arc observations, 2016 AMOS Conference technical papers - http://www.amostech.com/TechnicalPapers/2016.cfm Advanced Maui Optical and Space Surveillance Technologies Conference (AMOS)
The total number of active satellites, rocket bodies, and debris larger than 10 cm is currently about 20,000. Considering
all resident space objects larger than 1 cm this rises to an estimated minimum of 500,000 objects. Latest
generation sensor networks will be able to detect small-size objects, producing millions of observations per day. Due
to observability constraints it is likely that long gaps between observations will occur for small objects. This requires
to determine the space object (SO) orbit and to accurately describe the associated uncertainty when observations are
acquired on a single arc. The aim of this work is to revisit the classical least squares method taking advantage of the
high order Taylor expansions enabled by differential algebra. In particular, the high order expansion of the residuals
with respect to the state is used to implement an arbitrary order least squares solver, avoiding the typical approximations
of differential correction methods. In addition, the same expansions are used to accurately characterize the
confidence region of the solution, going beyond the classical Gaussian distributions. The properties and performances
of the proposed method are discussed using optical observations of objects in LEO, HEO, and GEO.
Lavagna M, Armellin R, Morselli A, Serpelloni E (2012) ASTREA: a Mars Lagrangian point base to serve asteroids belt cycling missions,
Armellin R, Brambilla A, Ercoli Finzi A, Lavagna M (2005) A distributed approach for space system formation operation autonomous planning and scheduling,
The paper presents a possible approach to the planning-scheduling to be faced by a space system formation to
accomplish specific mission goals with limited on-board resources. A multi-agent architecture is here selected to
cope with flexibility and reliability requirements typical for a system that must be operative in a hostile and not
completely known environment such the space environment is. A distributed versus a central architecture is here
preferred, and the communication protocol and negotiation mechanisms are investigated. The CSP problem is solved
both by applying to the STN scheme a Shortest Path algorithm and a Maximum Weighted Cliques technique for the
resources management. The applicative scenario here proposed consists of an Earth spacecraft formation devoted to
accomplish both single and globally shared activities by exploiting both local and global resources. Reconfiguration
maneuvers are included, to be accomplished by an electric propulsion module. That kind of operations make the set
of temporal constraints to be satisfied by the final schedule more complex as the search mechanism has to cope with
a nested optimal control problem. A dedicated light and fast algorithm to solve the control problem for the
reconfiguration maneuvers has been developed. Preliminary results according to the formation flying application are
Orbit perturbations are fundamental when analyzing the long-term evolution and stability of natural or artificial satellites. We propose the computation of transfer maps for repetitive dynamical systems as a novel approach to study the long-term evolution of satellite and space debris motion. We provide two examples of this technique, the evolution of high area-to-mass ratio spacecraft under solar radiation pressure and J2, and a sun-synchronous groundtrack repeating orbit with drag and J2. The results presented demonstrate the potentiality of the transfer maps for these problems. We furthermore compare this approach with averaging methods for the propagation of the orbital dynamics on the long-term, and suggest possibilities to combine differential algebra based methods with orbital elements averaging.
Valli M, Armellin R, Di Lizia P, Lavagna MR (2013) Nonlinear mapping of uncertainties in celestial mechanics, Journal of Guidance, Control, and Dynamics 36 (1) pp. 48-63
The problem of nonlinear uncertainty propagation represents a crucial issue in celestial mechanics. In this paper, a method for nonlinear propagation of uncertainty based on differential algebra is presented. Working in the differential algebra framework enables a general approach to nonlinear uncertainty propagation that can provide highly accurate estimate with low computational cost. The nonlinear mapping of the statistics is shown here, adopting the two-body problem as the working framework, including coordinate system transformations. The general feature of the proposed method is also demonstrated by presenting long-term integrations in complex dynamic systems, such as the n-body problem or the simplified general perturbation model.
Lavagna M, Armellin R, Bombelli R, Benvenuto R (2012) Debris removal mechanism based on tethered nets, Proceedings ? International Symposium on Artificial Intelligence, Robotics and Automation in Space i?SAIRAS 2012, 4?6 September, Turin, Italy
A multidisciplinary-multiobjective optimization of aerocapture maneuvers is presented. The proposed approach allows a detailed analysis of the coupling among vehicle's shape, trajectory control, and thermal protection system design. A set of simplified models are developed to address this analysis and a multiobjective particle swarm optimizer is adopted to obtain the set of Pareto optimal solutions. In order to deal with an unconstrained multiobjective optimization, a two-point boundary value problem is formulated to implicitly satisfy the constraints on the atmospheric exit conditions. The trajectories of the most promising solutions are further optimized in a more refined dynamical system by solving an optimal control problem using a direct multiple shooting transcription method. Furthermore, a more complete vehicle control is considered. All the simulations presented consider an aerocapture at Mars with a polar orbit of 200 km of altitude as target orbit.
Nonlinear filtering plays an important role in various space-related applications and especially in orbit determination and navigation problems. Differential algebraic techniques are here proposed as a valuable tool to reduce the computational burden of Monte Carlo based filtering algorithms, and in particular of the ensemble Kalman filter, without losing accuracy. The performance of the proposed filter is assessed on typical problems in spacecraft navigation. The cases of low estimation frequency and nonlinear measurements are also considered. Copyright© (2012) by the International Astronautical Federation.
Dowlat D, Armellin R, Lavagna M (2012) Lunar soft landing trajectory optimization in a 6DoF dynamical model, Proceedings of the 62nd International Astronautical Congress 2011 (IAC 2011) 7 pp. 5308-5318 International Astronautical Federation
An algorithm for the optimization of a lunar soft-landing trajectory is presented. A 6DoF modeling of the dynamics is adopted together with an accurate description of the Moon gravity field. The problem is faced as a direct optimization problem with the goal of obtaining a vertical landing whilst minimizing the overall fuel consumption. The descent trajectory is supposed to start from the periselenium of a low Moon orbit. Four optimization phases are considered. Each phase is characterized by a different set of optimization variables, constraints, and increasing level of complexity. In the first phase the thrust direction is optimized considering the translational motion of the lander only. Furthermore, no throttle capability is considered. In the second phase the thrust direction is fixed in the spacecraft body reference frame. The proper thrust orientation is obtained by optimizing the control torques supplied to the lander by the attitude sub-system. In the third phase the thrust magnitude is optimized too, and the constraint of landing on specific site is added. Furthermore, more restrictive constraints on the final velocities (linear and angular) are set. Finally, in the fourth phase a more accurate gravitational model of Moon that includes the main harmonics is considered. The algorithm is tested on a lunar landing trajectory in an area close to the Moon's equator, with a candidate spacecraft for Google Lunar X-prize.
In this work both reconfiguration and station keeping maneuvers of spacecraft flying in formation with low-thrust propulsion has been faced. The aim is the development of a real-time control strategy that reduces the propellant fraction required for these maneuvers. First a method to generate the optimal trajectories has been developed using a direct approach and a simplified model of the relative dynamics with respect to elliptic reference orbits, then the same method has been applied to control the maneuvers. Finally simulations considering a complete model have been carried out for different test cases.
A method for high-order treatment of uncertainties in preliminary orbit determination is presented. The observations consist in three couples of topocentric right ascensions and declinations at three observation epochs. The goal of preliminary orbit determination is to compute a trajectory that fits with the observations in two-body dynamics. The uncertainties of the observations are usually mapped to the phase space only when additional observations are available and a least squares fitting problem is set up. A method based on Taylor differential algebra for the analytical treatment of observation uncertainties is implemented. Taylor differential algebra allows for the efficient computation of the arbitrary order Taylor expansion of a sufficiently continuous multivariate function. This enables the mapping of the uncertainties from the observation space to the phase space as high-order multivariate Taylor polynomials. These maps can then be propagated forward in time to predict the observable set at successive epochs. This method can be suitably used to recover newly discovered objects when a scarce number of measurements is available. Simulated topocentric observations of asteroids on realistic orbits are used to assess the performances of the method.
The evolution of the hardware and the capabilities of the general computer algebra system
have supplied us with the possibility of developing an environment called
MathATESAT embedding in Mathematica, which is not linked to a Poisson series processor.
MathATESAT implements all the necessary tools to carry out high accuracy analytical
or semi-analytical theories in order to analyze the quantitative and qualitative behavior
of a dynamic system.
Massari M, Armellin R, Finzi AE (2005) Optimal trajectory generation and control for reconfiguration maneuvers of formation flying using low-thrust propulsion, Advances in the Astronautical Sciences 119 (Pt.1: Proceedings of the AAS/AIAA Space Flight Mec) pp. 2461-2474 American Astronautical Society
In this work the reconfiguration of spacecrafts flying in formation with low-thrust propulsion has been faced. The aim is the development of a control strategy that reduces the propellant use for reconfiguration maneuvers and that can be applied in real time. First a method to generate the optimal trajectory for the entire formation has been developed using a direct approach and a simplified model of the relative dynamics with respect to elliptic reference orbits, then the same method has been applied to control the maneuvers. Finally simulations considering a complete model have been carried out for different kind of reconfigurations.
A method for the nonlinear propagation of uncertainties in Celestial Mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis' close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis.
Mistry D, Armellin R (2016) The design and optimisation of end-of-life disposal manoeuvres for gnss spacecraft: The case of galileo, Proceedings of the 66th International Astronautical Congress, 2015 3 pp. 2187-2199
Nowadays there is international consensus that space activities must be managed to minimize debris generation and risk. The paper presents a method for the cnd-of-life (EoL) disposal of spacecraft in Medium Earth Orbit (MEO). The problem is formulated as a multiobjective optimization one, which is solved with an evolutionary algorithm. One or two impulses are design either to re-enter the spacecraft in Earth's atmosphere or to move them to a graveyard orbit. In the design of the re-entry the maximization of eccentricity build-up is achieved by using the minimization of the pericenter radius and disposal V as problem objectives. When the graveyard disposal is addressed, the long-Term probability of interfere with operative spacecraft is minimized together with the disposal V. To explore at the best the search space a semi-Analytical orbit propagator, which allows the propagation of the orbit motion for 100 years in few seconds, is adopted. Test cases for different navigation satellites are presented, focusing in particular on the Galileo constellation. The results are then used for a critical analysis on MEO missions' disposal and on the suitable definition of graveyard regions.
Optimal feedback control is classically based on linear approximations, whose accuracy drops off rapidly in highly nonlinear dynamics. Several nonlinear optimal feedback control strategies have appeared in recent years. Among them, differential algebraic techniques have been used to tackle nonlinearities by expanding the solution of the optimal control problem about a reference trajectory and reducing the computation of optimal feedback control laws to the evaluation of high order polynomials. However, the resulting high order method could not handle control saturation constraints, which remain a critical facet of nonlinear optimal feedback control. This work introduces the management of saturating actuators in the differential algebraic method. More specifically, the constraints are included in the optimal control problem formulation and differential algebra is used to expand the associated optimal bang-bang solution with respect to the initial and terminal conditions. Optimal feedback control laws for thrust direction and switching times are again computed by evaluating the resulting polynomials. Illustrative applications are presented in the frame of the optimal low-thrust transfer to asteroid 1996 FG3.
Lidtke A, Lewis HG, Armellin R (2014) A deterministic approach to active debris removal target selection,
Many decisions, with widespread economic, political and legal consequences, are being considered based on the concerns about the sustainability of spaceflight and space debris simulations that show that Active Debris Removal (ADR) may be necessary.
The debris environment predictions are affected by many sources of error, including low-accuracy ephemerides and propagators. This, together with the inherent unpredictability of e.g. solar activity or debris attitude, raises doubts about the ADR target-lists that are produced. Target selection is considered highly important, as removal of non-relevant objects will unnecessarily increase the overall mission cost .
One of the primary factors that should be used in ADR target selection is the accumulated collision probability of every object . To this end, a conjunction detection algorithm, based on the ?smart sieve? method, has been developed and utilised with an example snapshot of the public two-line element catalogue. Another algorithm was then applied to the identified conjunctions to estimate the maximum and true probabilities of collisions taking place.
Two target-lists were produced based on the ranking of the objects according to the probability they will take part in any collision over the simulated time window. These probabilities were computed using the maximum probability approach, which is time-invariant, and estimates of the true collision probability that were computed with covariance information.
The top-priority targets are compared, and the impacts of the data accuracy and its decay highlighted. General conclusions regarding the importance of Space Surveillance and Tracking for the purpose of ADR are drawn and a deterministic method for ADR target selection, which could reduce the number of ADR missions to be performed, is proposed
A novel ASI Lunar mission is here proposed by a task force of Ph.D. students. After 14 th January 2004 president G.W Bush's speech, a new input to space human exploration has been given. The Moon, thanks to nearness to Earth, is identified as an important test bed for all future human missions. The task force LUME mission has been designed to fit with Italian technological capabilities leaving it open anyway for international cooperation. Three main module are foreseen: a lunar low altitude polar orbiter, a lander near the "peak of the eternal light" and a rover. The polar orbiter is equipped with a complete suite of experiments for remote sensing observation (high resolution color camera, VIS-NIR imaging spectrometer, neutron and X spectrometers and SAR radar). This will provide a lunar surface map in high spatial resolution at different wavelengths: the orbiter payload will be used both to refine the selection of the landing site and to support the rover navigation. The lander will reach the region of "peak of the eternal light", located in the South Pole-Aitken Basin. This landing site has been selected for two main reasons: a) sun-light is always available to deliver the power useful to perform lander experiments and b) some easy-reachable and interesting craters are close to this region. The lander embark a sun powered ISRU plant to demonstrate O 2 extraction from lunar (ilmenite) soil and a robotic arm that can pick up lunar samples both from the soil and the rover. The nuclear powered rover is equipped with a drill system that, in the first phase of its mission, will deliver samples to be processed by the ISRU plant. In a second phase the rover will move to "de Gerlache" crater, identified as an attractive region to search for water ice. The rover drill includes an imaging VIS-NIR spectrometer dedicated to analyze the mineral composition and the water ice presence along the walls of the excavated hole. Both the orbiter and the lander will carry as payload two aquatic enclosed ecosystems (biospheres): these systems have been chosen as the best trade off between reduced requirements and easy data comprehension to evaluate space environment effects on life.
In this paper a differential algebra version of the gravity assist space pruning algorithm is presented. The use of differential algebraic techniques is proposed to overcome the two main drawbacks of the existing algorithm, i.e., the steep increase of the number of function evaluations with the number of planets involved in the transfer, and the use of a bounding procedure that relies on Lipschitzian tolerances. Differential algebra allows us to process boxes in place of grid points, and to substitute pointwise evaluations of the constraint functions with their Taylor expansions. Thanks to the particular instance of multi-gravity assist problems dealt with, all the planet-to-planet legs can be treated independently, and forward and backward constraining can be applied. The proposed method is applied to preprocess the search space of sample interplanetary transfers and it also serves as a stepping stone towards a fully rigorous treatment of the pruning process based on Taylor models.
Valli M, Armellin R, Di Lizia P, Lavagna MR (2012) Nonlinear management of uncertainties in celestial mechanics, Advances in the Astronautical Sciences, vol.143: Spaceflight Mechanics 2012 143 (3) pp. 2429-2446 Univelt Inc.
The problem of nonlinear uncertainty propagation represents a crucial issue in celestial mechanics. In this paper a method for nonlinear propagation of uncertainties based on differential algebra is presented. Working in the differential algebra framework enables a general approach to nonlinear uncertainty propagation that can provide high estimate accuracy with low computational burden. The nonlinear mapping of the statistics is here shown adopting the two-body problem as working framework, including coordinate system transformations. The general feature of the proposed method is also demonstrated by presenting long-term integrations in a complex dynamical framework, such as the n-body problem or the HANDE model.
Lunghi P, Armellin R, Di Lizia P, Lavagna M (2016) Semi-Analytical Adaptive Guidance Computation Based on Differential Algebra for Autonomous Planetary Landing, Spaceflight Mechanics volume 158: Proceedings of the 26th AAS/AIAA Space Flight Mechanics Meeting held February 14?18, 2016, Napa, California, U.S.A. 158 pp. 2003-2022 American Astronautical Society (AAS)
A novel algorithm for autonomous landing guidance computation is presented. Trajectory
is expressed in polynomial form of minimum order to satisfy a set of 17 boundary constraints,
depending on 2 parameters: time-of-flight and initial thrust magnitude. The consequent
control acceleration is expressed in terms of differential algebraic (DA) variables,
expanded around the point of the domain along the nominal trajectory followed at the retargeting
epoch. The DA representation of objective and constraints give additional information
about their sensitivity to variations of optimization variables, exploited to find the
desired fuel minimum solution (if it exists) with a very light computational effort, avoiding
less robust processes.
Di Lizia P, Armellin R, Bernelli Zazzera F, Jagasia R, Makino K, Berz M (2009) Validated integration of solar system dynamics,
The Earth orbits the Sun in a sort of cosmic shooting gallery, subject to impacts from comets and asteroids. It is only
fairly recently that we have come to appreciate that these impacts by asteroids and comets (often called Near Earth
Objects, or NEO) pose a significant hazard to life and property. Although the probability of the Earth being struck by a
large NEO is extremely small, the consequences of such a collision are so catastrophic that it is adviseable to assess the
nature of the threat and prepare to deal with it.
One of the major issues in determining whether an asteroid can be dangerous for the Earth is given by the uncertainties
in the determination of its position and velocity. Important studies arose from the previous issue, which dealt with the
problem of getting accurate uncertainty estimates of the state of orbiting objects by means of several approaches, among
which statistical theory has showed important results [1,2,3]. Recently, several tools and techniques have been
developed for the robust prediction of Earth close encounters and for the identification of possible impacts of NEO with
the Earth [4,5,6]. However, these methods might suffer from being either not sufficiently accurate when relying on
simplifications (e.g., linear approximations) or computationally intensive when based on several integration runs (e.g.,
the Monte Carlo approach). Moreover, the standard integration schemes are affected by numerical integration errors,
which might unacceptably accumulate during the integration, especially when long term integrations are performed.
The resulting inaccuracies might strongly affect the validity of the results, precluding the use of such integrators.
The necessity of solving these problems brought about a strong interest in self validated integration methods , which
are based on the use of interval analysis. Interval analysis was originally formalized by Moore in 1966 . The main
idea beneath this theory is the substitution of real numbers with intervals of real numbers; consequently, interval
arithmetic and analysis are developed in order to operate on the set of interval numbers in place of the classical analysis
of real numbers. This turned out to be an effective tool for error and uncertainty propagation, as both the numerical
errors and the uncertainties can be bounded by means of intervals, which are rigorously propagated in the computation
by operating on them using interval analysis.
Unfortunately, the naïve appli
Current approaches to uncertainty propagation in astrodynamics mainly refer to linearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails when the non-linearities of the dynamics prohibit good convergence of the Taylor expansion in one or more directions. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial expansion of the current state is split into two polynomials whenever its truncation error reaches a predefined threshold. The resulting set of polynomials accurately tracks uncertainties, even in highly nonlinear dynamics. The method is tested on the propagation of (99942) Apophis post-encounter motion.
The aero-gravity assist maneuver is here proposed as a tool to improve the efficiency of the gravity assist as, thanks to the interaction with the planetary atmospheres, the angular deviation of the velocity vector can be definitely increased. Even though the drag reduces the spacecraft velocity, the overall Å gain could be remarkable whenever a high lift-to-drag vehicle is supposed to fly. Earlier studies offer simplified approaches according to both the dynamics modeling and the atmospheric trajectory constraints. In this paper a 3D dynamical model is adopted and a more realistic L/D performance for the hypersonic vehicle is assumed. Some relevant aspects related to the multidisciplinary design have been considered such as heating rates and structural loads bounding. Comparisons between in and out of plane maneuvering have been performed by assuming, as control variables, either the angle of attack or the bank angle, respectively. The optimal control problem has been solved by selecting a direct method approach. The dynamics has been transcribed into a set of non-linear constraints and the arising non-linear programming problem has been solved with a sequential quadratic programming solver. To gain the global optimum convergence the initial guess has been supplied by solving the same problem by a direct shooting technique and a genetic optimizer. © Springer Science+Business Media B.V. 2006.
Di Lizia P, Armellin R, Bernelli-Zazzera F (2013) Robust optimal control of low-thrust interplanetary transfers, Advances in the Astronautical Sciences vol.148: Proceedings of the 23rd AAS/AIAA Space Flight Mechanics Meeting held February 10?14, 2013, Kauai, Hawaii 148 pp. 803-822 American Astronautical Society
Continuous-thrust orbit transfers are designed by solving an optimal control problem that minimizes fuel consumption while satisfying mission constraints. The optimal control problem is usually solved in nominal conditions: at the design stage, the dynamics modeling is supposed to exactly represent the reality. An algorithm to include uncertain parameters and boundary conditions is presented. This is based on the high-order expansion of the solution of the two-point boundary value problem associated to the optimal control problem with respect to uncertainties. Illustrative applications are presented in the frame of the optimal low-thrust transfer to asteroid 1996 FG3.
Bombelli A, Benvenuto R, Carta R, Lavagna M, Armellin R (2012) Optimal design of a net-shaped space debris removal system,
In this chapter, differential algebra is used to globally optimize multigravity assist interplanetary trajectories with deep space maneuvers. A search space pruning procedure is adopted, and the trajectory design is decomposed into a sequence of sub-problems. As far as differential algebra is used, the objective function and the constraints are represented by Taylor series of the design variables over boxes in which the search space is divided. Thanks to the polynomial representation of the function and the constraints, a coarse grid can be used, and an efficient design space pruning is performed. The manipulation of the polynomials eases the subsequent local optimization process, so avoiding the use of stochastic optimizers. These aspects, along with the efficient management of the list of boxes, make differential algebra a powerful tool to design multi-gravity assist transfers including deep-space maneuvers.
Armellin R, Di Lizia P, Bernelli Zazzera F, Makino K, Berz M (2009) Apophis encounter 2029: differential algebra and Taylor model approaches, Proceedings of the 1st IAA Planetary Defense Conference: Protecting the Earth from Asteroids, ESA Proceedings, Grenada (Spain), April 26-30, 2009
Orbit uncertainty propagation usually requires linearized propagation models [1?3] or full nonlinear Monte Carlo
simulations . The linear assumption simplifies the problem, but fails to characterize trajectory statistics when the
system is highly nonlinear or when mapped over a long time period. On the other hand, Monte Carlo simulations
provide true trajectory statistics, but are computationally intensive. The tools currently used for the robust detection and
prediction of planetary encounters and potential impacts with Near Earth Objects (NEO) are based on these two
techinques [5?7], and thus suffer the same limitations. A different approach to orbit uncertainty propagation has been
discussed by Junkins et al. [8,9], in which the effect of the coordinate system on the propagated statistics is thoroughly
analyzed; however, the propagation method was based on the linear assumption and the system nonlinearity was not
incorporated in the mapping. An alternate way to analyze trajectory statistics by incorporating higher-order Taylor
series terms that describe localized nonlinear motion was proposed by Park and Scheeres . Their appoach is based
on proving the integral invariance of the probability density function via solutions of the Fokker?Planck equations for
diffusionless systems, and by combining this result with the nonlinear state propagation to derive an analytic
representation of the nonlinear uncertainty propagation. This method is limited to systems derived from a single
Differential algebraic (DA) techniques are proposed as a valuable tool to develop alternative approaches to tackle the
previous tasks. Differential algebra provides the tools to compute the derivatives of functions within a computer
environment [11?13]. More specifically, by substituting the classical implementation of real algebra with the
implementation of a new algebra of Taylor polynomials, any function f of v variables is expanded into its Taylor series
up to an arbitrary order n. This has an important consequence when the numerical integration of an ordinary differential
equation (ODE) is performed by means of an arbitrary integration scheme. Any explicit integration scheme is based on
algebraic operations, involving the evaluations of the ODE right hand side at several integration points. Therefore,
carrying out all the evaluation in the DA framework allows differential algebra to compute the arbitrary order expansion
of the flow of a general ODE initial val
A method intended to assess the occurrence of impacts between satellites and space debris is presented. The method is based on the computation of the minimum orbit intersection Distance (MOID) between two perturbed orbits. The MOID is obtained by means of a global optimization, using a global optimizer based on Taylor models. The position of the orbiting objects is described through analytical solutions that take into account zonal harmonics and atmospheric drag. The global optimizer searches the global minimum of the square distance between the two orbits, that is a function of the two true anomalies and time. The optimization is capable of providing tight enclosures of the global minimum. The method is applied to the case of a Sun-synchronous orbit and an equatorial elliptic orbit. Due to the effect of the Earth zonal harmonics, the orbital plane of the Sun-synchronous orbit performs a complete rotation around Earth's North Pole direction. The relative geometry of the two orbits is such that four intersections occur within one year window. A second test-case demonstrates the ability to predict the effect of drag perturbations. Two LEO orbits are considered, one of which is strongly influenced by atmospheric drag. The decrease of semi-major axis leads to an intersection between the two orbits in the considered time span. The method is capable to correctly identify the intersections between two orbits and, thus, the conditions in which close approaches between satellites and debris occur. The method is intended to perform a fast screening of all possible combinations of orbiting object, since the use of analytical theory cuts down simulation time.
Armellin R, Di Lizia P (2016) Probabilistic initial orbit determination, Proceedings of the 26th AAS/AIAA Space Flight Mechanics Meeting held February 14?18, 2016, Napa, California, U.S.A.
American Astronautical Society (AAS)
Future space surveillance requires dealing with uncertainties directly in the initial orbit determination phase. We propose an approach based on Taylor differential algebra to both solve the initial orbit determination (IOD) problem and to map uncertainties from the observables space into the orbital elements space. This is achieved by approximating in Taylor series the general formula for pdf mapping through nonlinear transformations. In this way the mapping is obtained in an elegant and general fashion. The proposed approach is applied to both angles-only and two position vectors IOD for objects in LEO and GEO.
A high order method to quickly assess the effect that uncertainties produce on orbital conjunctions through a numerical high-fidelity propagator is presented. In particular, the dependency of time and distance of closest approach to initial uncertainties on position and velocity of both objects involved in a conjunction is studied. The approach relies on a numerical integration based on differential algebraic techniques and a high-order algorithm that expands the time and distance of closest approach in Taylor series with respect to relevant uncertainties. The modeled perturbations are atmospheric drag, using NRLMSISE-00 air density model, solar radiation pressure with shadow, third body perturbation using JPL's DE405 ephemeris, and EGM2008 gravity model. The polynomial approximation of the final position is used as an input to compute analytically the expansion of time and distance of closest approach. As a result, the analysis of a close encounter can be performed through fast, multiple evaluations of Taylor polynomials. Test cases with objects ranging from LEO to GEO regimes are considered to assess the performances and the accuracy of the proposed method.
This work investigates end of life disposal options for libration point orbit missions. Three different options are presented: the first one considers spacecraft's re-entry in Earth's atmosphere, the second one concerns the impact on the Moon, whereas the third one consists in the injection of the spacecraft into a heliocentric graveyard orbit. The disposal design is formulated as a multi-objective optimization problem in order to take into account other goals in addition to propellant consumption minimization. The disposal of Gaia mission is used as test case throughout the paper.
Guzetti D, Armellin R, Lavagna M (2012) Coupling attitude and orbital motion of extended bodies in the restricted circular 3-body problem: a novel study on effects and possible exploitations, Proceedings of the 63rd International Astronautical Congress, Naples, Italy pp. 5715-5728 International Astronautical Federation
The rigorous solution of a generic impulsive planet-to-planet transfer by means of a Taylor-model-based global optimizer is presented. Although a planet-to-planet transfer represents the simplest case of interplanetary transfer, its formulation and solution is a challenging task when the rigorous global optimum is sought. A customized ephemeris function is derived from JPL DE405 to allow the Taylor-model evaluation of planets positions and velocities. Furthermore, the validated solution of Lambert's problem is addressed for the rigorous computation of transfer fuel consumption. The optimization problem, which consists in finding the optimal launch and transfer time to minimize the required fuel mass, is complex due to the abundance of local minima and relatively high search-space dimension. Its rigorous solution by means of the Taylor-model-based global optimizer COSY-GO is presented considering Earth-Mars and Earth-Venus transfers as test cases.
Di Mauro G, Di Lizia P, Armellin R, Lavagna M (2012) A novel nonlinear control approach for rendezvous and docking maneuvering, Proceedings of the 63rd International Astronautical Congress (IAC), 2012 7 pp. 5357-5365 International Astronautical Federation
Nowadays autonomous rendezvous and docking (ARD) operations represent a crucial technique for
several space missions which involve either in orbit assembly of numerous modules or serving/refueling
operations; both around Earth and interplanetary scenarios, manned and unmanned vehicles may ask
for such a technique. In order to obtain the nal docking interface conditions, translational and attitude
constraints must be satis ed during a rendezvous and docking maneuver; therefore, the control scheme
has to simultaneously tune the relative position and velocity between the two satellites and adjust the
chaser spacecraft's orientation with respect to the target port. Since dynamic equations which describe
the relative satellites pose are nonlinear, many nonlinear control methodologies have been investigated
during last decade: Terui has shown the e ectiveness of the sliding mode control technique to control
both position and attitude for proximity
ight around a tumbling target; Kim et al. have proposed
nonlinear backstepping control method to solve the spacecraft slew manoeuvre problem. One of the
highly promising and rapidly emerging methodologies for designing nonlinear controllers is the state-
dependent Riccati equation (SDRE) approach, originally proposed by Pearson and Burghart and then
described in details by Cloutier, Hammett and Beeler. This approach manipulates the governing dynamic
equations into a pseudo-linear non-unique (SDC parameterization) form in which system matrices are
given as a function of the current state and minimizes a quadratic-like performance index. Then, a sub-
optimal control law is obtained by online solution of Algebraic Riccati Equation (ARE). Even if the SDRE
method represents a valid option to solve the nonlinear control problem related to ARD manoeuvring, it
is prone to high computational costs due to the online solution of ARE. In this paper, a new approximated
SDRE solution to solve ARD manoeuvring problem based on the di erential algebra (DA) exploitation
is proposed. Di erential algebraic techniques allow for the e cient computation of the arbitrary order
Taylor expansion of a su ciently continuous multivariate function in a computer environment with a xed
resource demand. A DA-based algorithm is here presented to compute a high order Taylor expansion of
the SDRE solution with respect to the state vectors around a reference trajectory. The computation of
the next SDRE solution is then reduced to the mere evaluation of
Current approaches to uncertainty propagation in astrodynamics mainly refer to linearize models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationally intensive. Differential algebra has already proven to be an efficient compromise by replacing thousands of pointwise integrations of Monte Carlo runs with the fast evaluation of the arbitrary order Taylor expansion of the flow of the dynamics. However, the current implementation of the DA-based high-order uncertainty propagator fails in highly nonlinear dynamics or long term propagation. We solve this issue by introducing automatic domain splitting. During propagation, the polynomial of the current state is split in two polynomials when its accuracy reaches a given threshold. The resulting polynomials accurately track uncertainties, even in highly nonlinear dynamics and long term propagations. Furthermore, valuable additional information about the dynamical system is available from the pattern in which those automatic splits occur. From this pattern it is immediately visible where the system behaves chaotically and where its evolution is smooth. Furthermore, it is possible to deduce the behavior of the system for each region, yielding further insight into the dynamics. In this work, the method is applied to the analysis of an end-of-life disposal trajectory of the INTEGRAL spacecraft.
Two-point boundary value problems appear frequently in space trajectory design. A remarkable example is represented by the Lambert's problem, where the conic arc linking two fixed positions in space in a given time is to be characterized in the frame of the two-body problem. Classical methods to numerically solve these problems rely on iterative procedures, which turn out to be computationally intensive in case of lack of good first guesses for the solution. An algorithm to obtain the high order expansion of the solution of a two-point boundary value problem is presented in this paper. The classical iterative procedures are applied to identify a reference solution. Then, differential algebra is used to expand the solution of the problem around the achieved one. Consequently, the computation of new solutions in a relatively large neighborhood of the reference one is reduced to the simple evaluation of polynomials. The performances of the method are assessed by addressing typical applications in the field of spacecraft dynamics, such as the identification of halo orbits and the design of aerocapture maneuvers.
Morselli A, Armellin R, Di Lizia P, Bernelli Zazzera F (2015) A high order method for orbital conjunctions analysis: Monte Carlo collision probability computation, Advances in Space Research 55 (1) pp. 311-333
Published by Elsevier Ltd. All rights reserved.Three methods for the computation of the probability of collision between two space objects are presented. These methods are based on the high order Taylor expansion of the time of closest approach (TCA) and distance of closest approach (DCA) of the two orbiting objects with respect to their initial conditions. The identification of close approaches is first addressed using the nominal objects states. When a close approach is identified, the dependence of the TCA and DCA on the uncertainties in the initial states is efficiently computed with differential algebra (DA) techniques. In the first method the collision probability is estimated via fast DA-based Monte Carlo simulation, in which, for each pair of virtual objects, the DCA is obtained via the fast evaluation of its Taylor expansion. The second and the third methods are the DA version of Line Sampling and Subset Simulation algorithms, respectively. These are introduced to further improve the efficiency and accuracy of Monte Carlo collision probability computation, in particular for cases of very low collision probabilities. The performances of the methods are assessed on orbital conjunctions occurring in different orbital regimes and dynamical models. The probabilities obtained and the associated computational times are compared against standard (i.e. not DA-based) version of the algorithms and analytical methods. The dependence of the collision probability on the initial orbital state covariance is investigated as well.
Lunghi P, Lavagna M, Armellin R (2014) Adaptive semi-analytical guidance for autonomous planetary landing, Proceedings of the 64th International Astronautical Congress 7 pp. 5103-5113
Hazard Detection and Avoidance (HDA) is a key technology for a safe landing in future planetary missions. HDA systems require the capability of modify the landing trajectory with high precision almost to the touchdown. An adaptive guidance algorithm for planetary landing that updates the trajectory to the surface by means of a minimum fuel optimal control problem solving is proposed. A semi-analytical approach is adopted. The trajectory is expressed in a polynomial form of minimum order to satisfy a set of boundary constraints derived from initial and final states and attitude requirements. By imposing boundary conditions, a fully determined guidance profile is obtained, function of only two parameters: time-of-flight and initial thrust magnitude. The optimal guidance computation is reduced to the determination of these parameters, according to additional path constraints due to the actual lander architecture: available thrust and control torques, visibility of the landing site, and other additional constraint not implicitly satisfied by the polynomial formulation. Solution is achieved with a simple two-stage compass search algorithm: the algorithm firstly finds a feasible solution; whenever detected, it keeps solving for the optimum; nonlinear constraints are evaluated numerically, by pseudospectral methods. Results on different scenarios for a Moon landing mission are shown and discussed to highlight the effectiveness of the proposed algorithm and its sensitivity to the navigation errors.
Armellin R, Di Lizia P, Morselli A, Valli M, Lavagna M, Bernelli-Zazzera F, Berz M (2012) High order algorithms for the management of uncertainties with applications in space situational awareness,
Space situational awareness program, for both
NEO and debris segments, have to face the
challenging problem of accurately managing uncertainties
in highly nonlinear dynamical environments.
The uncertainties affect all the
main phases necessary for the successful realization
of the program; i.e., orbital determination,
ephemeris prediction, collision probability computation,
and collision avoidancemaneuver planning
and execution. Since the amount of data
that must be processed is huge, efficient methods
for the management of uncertainties are required.
Differential algebraic (DA) techniques
can represent a valuable tool to address this tasks.
Differential algebra supplies the tools to compute
the derivatives of functions within a computer
environment. This technique allows for the efficient
computation of high-order expansions of
the flow of ordinary differential equations (with
respect to initial conditions and/or model parameters)
and the approximation of the solution manifold
of implicit equations in Taylor series. These
two features constitute the building blocks of a
set new algorithms for the nonlinear and efficient
management of uncertainties. Applications to
1) angles-only preliminary orbit determination 2)
propagation of orbital dynamics 3) nonlinear filtering
4) space conjunction prediction 5) robust
optimal control are presented to prove the efficiency
of DA based algorithms.
The study of orbital conjunctions between space bodies is of fundamental importance in space situational awareness programs. The identification of potentially dangerous conjunctions, either between Near Earth Objects and our planet, or space debris and operative spacecraft, is most commonly done by looking at the distance between the objects in a given time window. A method based on Taylor models and Taylor differential algebra is presented to compute the time and distance of closest approach and to assess the effect that uncertainties on initial orbital parameters produce on these quantities.
Armellin R, Di Lizia P, Makino K, Berz M (2008) Rigorous global optimization of impulsive planet-to-planet transfers,
Abstract. The rigorous solution of a generic impulsive planet-to-planet
transfer by means of a Taylor Model based global optimizer is presented.
Although a planet-to-planet transfer represents the simplest case of in-
terplanetary transfer, its formulation and solution is a challenging task
as far as the rigorous global optimum is sought. A customized ephemeris
function is derived from JPL DE405 to allow the Taylor Model evalu-
ation of planets? positions and velocities. Furthermore, the validated
solution of Lambert?s problem is addressed for the rigorous computa-
tion of transfer fuel consumption. The optimization problem, which
consists in finding the optimal launch and transfer time to minimize the
required fuel mass, is complex due to the abundance of local minima and
relatively high search space dimension. Its rigorous solution by means
of COSY-GO is presented considering Earth?Mars and Earth?Venus
transfers as test cases.
Armellin R, Di Lizia P, Di Mauro G, Rasotto M, Landgraf M (2014) Disposal strategies for spacecraft in Lagrangian Point Orbits, Advances in the Astronautical Sciences Series, Volume 152: SpaceFlight Mechanics 2014 152 (4) pp. 1731-1750 Univelt Inc.
This work presents three different strategies for the disposal of Lagrangian Point Orbit (LPO) missions that have been analyzed in the frame of the European Space Agency study "End-of-life disposal concepts for Lagrange-points and HEO missions". The first strategy analyzes a Moon impact scenario, the second one a reentry in Earth's atmosphere, whereas the third concerns the injection into heliocentric graveyard orbits. For Moon impact and Earth's reentry an optimization problem is set up to minimize the propellant consumption while satisfying constraints on terminal position. For the graveyard, two methods are proposed: the first is a fully numerical approach based on the solution of an optimization problem via genetic algorithms, whereas the second one is based on the Jacobi constant and on a tangential disposal maneuver designed to close the Hill's regions. In this paper, solutions compatible with mission constraints are presented for SoHO and GAIA.
A novel method to compute all critical points of the distance function between two Keplerian orbits (either bounded or unbounded) with a common focus is presented. The problem is attacked as a global optimization problem, solved by a rigorous global optimizer based on Taylor models. Thus, thigh enclosures of the stationary points are obtained. The embedded capability of the method of delivering high-order Taylor expansions is then used to analyze how uncertain orbital parameters affect the position of the stationary points and the associated distance values. Sample orbital sets and Apophis asteroid are used as test cases.
Massari M, Armellin R, Di Lizia P, Topputo F (2007) Reaching NEOs: solution for the second global trajectory optimization competition, Proceedings of XIX Congresso Nazionale AIDAA
The second global trajectory optimization competition was organized by the Outer Planets Mission Analysis Group
of the Jet Propulsion Laboratory and was announced in October 2006. The competition objective was to find the
global optimal low-thrust trajectory, maximizing a complex objective function over a huge search space. The
problem was a multiple asteroid rendezvous: a trajectory had to be designed for a low-thrust spacecraft which
departs from the Earth and subsequently performs a rendezvous with one asteroid chosen within each of four
defined groups of asteroids. In this paper we present the results obtained by the Aerospace Engineering Department
of Politecnico di Milano. The problem has been approached by means of a two-phase solution process. The first
phase aims at determining a preliminary solution close to the global optimum that is refined in the second phase
through a local optimization. In this work the two-phase approach is presented together with the obtained solution.
Morselli A, Armellin R, Di Lizia P, Bernelli-Zazzera F (2013) Computing collision probability using differential algebra and advanced monte carlo methods, Proceedings of the 63rd International Astronautical Congress 2012 (IAC 2012) 3 pp. 2311-2324
A method for the computation of the probability of collision between two space object is pre- sent.ed. The method is based on the Taylor expansion of the time of close approach (TCA) and distance of close approach (DCA) of the two orbiting objects. These quantities are computed during the conjunction detection phase by means of a global optimizer based on Differential Al-gebra, which is implemented in the language COSY-lnfinity. When a close approach is identified, TCA and DC A nominal values are computed. Subsequently, analytical expansions with respect to uncertainties in the initial states are obtained by means of Differential Algebra. The collision probability is then computed via Monte Carlo simulation, sampling values of initial position and velocity according to their estimated uncertainties. The newr value of DCA for the couple of virtual objects is computed evaluating its Taylor polynomial, using the sampled deviations from the nominal initial state. The minimum distance is then compared with the collision threshold, that is the diameter of the sphere that envelopes the two spacecraft. To improve the efficiency and accuracy of the method advanced Monte Carlo Markov Chain techniques are employed. The presented test cases considers conjunctions occurring in LEO, MEO and GEO. For each test case the collision probabilities are computed with both standard and advanced Monte Carlo methods. The computed probabilities and the associated computational times are then compared. The method is suitable for a wide range of orbits since no simplifications of the conjunction event are made, hence it can be applied to geosynchronous orbits, where relative velocity is lower. The effects of main orbital perturbations are accounted for in the computation of the relative distance.
Lidtke AA, Gondelach DJ, Armellin R, Colombo C, Lewis HG, Funke Q, Flohrer T (2016) Processing two line element sets to facilitate re-entry prediction of spent rocket bodies from geostationary transfer orbit,
Predicting the re-entry of space objects enables the risk they pose to the ground population to be managed. The more accurate the re-entry forecast, the more cost-efficient risk mitigation measures can be put in place. However, at present, the only publicly available ephemerides (two line element sets, TLEs) should not be used for accurate re-entry prediction directly. They may contain erroneous state vectors, which need to be filtered out. Also, the object?s physical parameters (ballistic and solar radiation pressure coefficients) need to be estimated to enable accurate propagation. These estimates are only valid between events that change object?s physical properties, e.g. collisions and fragmentations. Thus, these events need to be identified amongst the TLEs. This paper presents the TLE analysis methodology, which enables outlying TLEs and space events to be identified. It is then demonstrated how various TLE filtering stages improve the accuracy of the TLE-based re-entry prediction.
Space trajectory design always requires the solution of an optimal control problem in order
to maximize the payload launch-mass ratio while achieving the primary mission goals. A certain level
of approximation always characterizes the dynamical models adopted to perform the design process. Furthermore the state identification is usually affected by navigation errors. Thus, after the nominal optimal
solution is computed, a control strategy that assures the execution of mission goals in the real scenario
must be implemented. In this frame differential algebraic techniques are here proposed as an effective alternative tool to design the guidance law. By using differential algebra the final state dependency on initial
conditions, environmental and control parameters is represented by high order Taylor series expansions.
The mission constraints can then be solved to high order using a so-called high order partial inversion of
the polynomial relationship for every admissible uncertainty. The control strategy is eventually reduced to
a simple function evaluation. The performances of the proposed methods are assessed by two examples of
space mission trajectory design: a continuous propelled Earth-Mars transfer and an aerocapture maneuver
Armellin R, Topputo F (2005) An optimal h6 scheme for solving TPBVP in astrodynamics,
The present paper presents an accurate scheme for the solution of boundary
value problems with two-point nonlinear boundary conditions. The proposed scheme
is a linear multi-point method of sixth-order accuracy successfully used in uid
dynamics and here implemented for the rst time in astrodynamics applications.
It is an optimal scheme since a discretization molecule made up of just four grid
points assures an h6 order of accuracy. This kind of discretization allows to attain
an accuracy beyond the rst Dahlquist's stability barrier and simultaneously has
a simple formulation and numerical e ciency. Astrodynamics applications concern
the computation of libration point halo orbits, in the restricted three- and four-body
models, and the design of an optimal control strategy for a low thrust libration point
Guzzetti D, Lavagna M, Armellin R (2012) Invariant manifolds to design scientific operative orbits in the Pluto-Charon binary system, Advances in the Astronautical Sciences, vol.143: Spaceflight Mechanics 2012 143 (2) pp. 503-520 Univelt Inc.
Feasibility of operative orbits in the Pluto-Charon system has been investigated in this work. Given that currently only the New Horizon NASA mission will perform a quick fly-by of Pluto-Charon, the chance to close a spacecraft in orbit around the system would represent a significant add-on in the science knowledge domain and an interesting challenge from the flight dynamics perspective. A R3BP coupled with the invariant manifolds are the main tools here exploited to manage the trajectories design; possible itineraries and strategies, that can meet the requirements of costs minimization, long operative life and adequate coverage of the surfaces, are proposed.
Di Mauro G, Armellin R, Rasotto M, Di Lizia P, Funke Q, Flohrer T (2016) Design of optimal observation strategy for re-entry prediction improvement of GTOs upper stage,
In this paper, an automatic approach to design the observation strategy of spent upper stage moving on GTOs is presented. More specifically, the design is formulated as a multi-objective optimization problem solved by means of a multi-objective genetic algorithm (MOGA). This approach allows minimizing both the number of total measurements required to detect the object and the error on re-entry prediction. Within the optimization process a nonlinear OD algorithm is run to determine the estimates of both initial state and model parameters and the associated covariance matrix. The Nonlinear Least Square Estimator (NLSE) technique is implemented, exploiting the differential algebra framework for Jacobian matrix computation in order to reduce the computational effort related to OD problem solution. Finally, the software tool IRIS is developed to accurately simulate the observation campaign based on geometry and constraints of existing sensors currently available to the European Space Agency (ESA). Numerical simulations are performed to demonstrate the efficiency of the proposed approach.
Space debris simulations, e.g. those performed by the Inter-Agency Debris Coordination Committee (Liou et al., 2013), showed that the number of objects in orbit is likely to increase. This study analyses the uncertainty in the results of space debris simulations performed using semi-stochastic models that necessitate the use of Monte Carlo simulations, which are often used by the Inter-Agency Debris Coordination Committee, amongst other studies. Statistics of the possible numbers of objects in orbit and collisions over the next 200 years are generated for the ?mitigation only? scenario using a sample of 25,000 Monte Carlo runs. Bootstraps on the mean, median, variance, skewness and kurtosis of these distributions are performed. It is shown that the distribution of the objects predicted to be on-orbit becomes log-normal as collisions occur, and that Monte Carlo samples larger than traditionally used are needed to capture the debris simulation uncertainty.
The focus of this paper is the design and station keeping of repeat-groundtrack orbits for Sun-synchronous satellite. A method to compute the semimajor axis of the orbit is presented together with a station-keeping strategy to compensate for the perturbation due to the atmospheric drag. The results show that the nodal period converges gradually with the increase of the order used in the zonal perturbations up to J15. A differential correction algorithm is performed to obtain the nominal semimajor axis of the reference orbit from the inputs of the desired nodal period, eccentricity, inclination and argument of perigee. To keep the satellite in the proximity of the repeat-groundtrack condition, a practical orbit maintenance strategy is proposed in the presence of errors in the orbital measurements and control, as well as in the estimation of the semimajor axis decay rate. The performance of the maintenance strategy is assessed via the Monte Carlo simulation and the validation in a high fidelity model. Numerical simulations substantiate the validity of proposed mean-elements-based orbit maintenance strategy for repeat-groundtrack orbits.
This paper introduces and combines for the first time two techniques to allow long-term density propagation in astrodynamics. First, we introduce an efficient method for the propagation of phase space densities based on Diµerential Algebra (DA) techniques. Second, this DA density propagator is used in combination with a DA implementation of the averaged orbital dynamics through semi-analytical methods. This approach combines the power of orbit averaging with the efficiency of DA techniques. While the DA-based method for the propagation of densities introduced in this paper is independent of the dynamical system under consideration, the particular combination of DA techniques with averaged equations of motion yields a fast and accurate technique to propagate large clouds of initial conditions and their associated probability density functions very efficiently for long time. This enables the study of the long-term behavior of particles subjected to the given dynamics. To demonstrate the eµectiveness of the proposed approach, the evolution of a cloud of high area-to-mass objects in Medium Earth Orbit is reproduced considering the eµects of solar radiation pressure, the Earth?s oblateness and luni-solar perturbations. The method can easily propagate 10, 000 random fragments and their density for 1 year within a few seconds on a common desktop PC.
Future space surveillance requires dealing with uncertainties directly in the initial orbit determination phase. We propose an approach based on Taylor differential algebra to both solve the initial orbit determination (IOD) problem and to map uncertainties from the observables space into the orbital elements space. This is achieved by approximating in Taylor series the general formula for probability density function (pdf) mapping through nonlinear transformations. In this way the mapping is obtained in an elegant and general fashion. The proposed approach is applied to both angles-only and two position vectors IOD for objects in LEO and GEO.
Efficient long-term propagation of orbits is needed for e.g. the design of disposal orbits and analysis of their stability. Semi-analytical methods are suited for this as they combine accuracy and efficiency. However, the semi-analytical modelling of non-conservative forces is challenging and in general numerical quadrature is required to accurately average their effects, which reduces the efficiency of semianalytical propagation. In this work we apply Differential Algebra (DA) for efficient evaluation of the mean element rates due to drag. The effect of drag is computed numerically in the DA arithmetic such that in subsequent integration steps the drag can be calculated by only evaluating a DA expansion. The method is tested for decaying low Earth and geostationary transfer orbits and it is shown that the method can provide accurate propagation with reduced computation time with respect to nominal semi-analytical and numerical propagation. Furthermore, the semi-analytical propagator is entirely implemented in DA to enable higherorder expansion of the flow that can be used for efficient propagation of initial conditions. The approach is applied to expand the evolution of a Galileo disposal orbit. The results show a large validity domain of the expansion which represents a promising result for the application of the method for e.g. stability analysis.
The object of this paper is the initial orbit determination of an orbit set compatible with an observational arc by means of differential algebra. The initial estimate is retrieved as a truncated power series expanded with respect to the uncertainties in the measurements. The analysis of the region is performed with the automatic domain splitting, that splits the orbit set into two or more regions defined by just as many Taylor expansions when the estimated truncation error introduced by the truncated power series exceeds a certain tolerance. A comparison between the proposed initial orbit determination approach and alternative methods from the literature is included to show the improvements achieved by exploiting accuracy information using differential algebra. The goal of the description of the initial orbit determination solution as orbit set is to propagate several independent orbit set to a common epoch and analyze them to decide whether they?re correlated or not. Initial results for the linkage of two independent observations are also included.
The uncertainty region associated with short arcs is typically large, making the initialization of orbit estimators a challenging task. In this work we propose a method to reduce the size of the uncertainty region using automatic domain pruning. The initial orbit and its confidence region are obtained by using a differential algebra-based initial orbit determination algorithm and a least squares algorithm. New measurements are used to reduce the size of the confidence region by retaining only those portions of the domain in which the likelihood is above a certain threshold.
The solution of multiple-revolution perturbed Lambert problems is a challenging task due
to the high sensitivity of the final state to variations of the initial velocity. In this work two
different solvers based on high order Taylor expansions and an analytical solution of the J2
problem are presented. In addition, an iteration-less procedure is developed to refine the
solutions in a dynamical model that includes J2 ? J4 perturbations. The properties of the
proposed approached are tested against transfers with hundreds of revolutions including those
required to solve the Global Trajectory Optimisation Competition 9.
Spent rocket bodies in geostationary transfer orbit (GTO) pose
impact risks to the Earth's surface when they re-enter the Earth's at-
mosphere. To mitigate these risks, re-entry prediction of GTO rocket
bodies is required. In this paper, the re-entry prediction of rocket bod-
ies in eccentric orbits based on only Two-Line Element (TLE) data
and using only ballistic coefficient (BC) estimation is assessed. The
TLEs are preprocessed to filter out outliers and the BC is estimated
using only semi-major axis data. The BC estimation and re-entry pre-
diction accuracy are analyzed by performing predictions for 101 rocket
bodies initially in GTO and comparing with the actual re-entry epoch
at different times before re-entry. Predictions using a single and mul-
tiple BC estimates and using state estimation by orbit determination
are quantitatively compared with each other for the 101 upper stages.
Nowadays there is international consensus that space activities must be managed to minimize debris generation and risk. The paper presents a method for the end-of-life (EoL) disposal of spacecraft in Medium Earth Orbit (MEO). The problem is formulated as a multiobjective optimisation one, which is solved with an evolutionary algorithm. An impulsive manoeuvre is optimised to reenter the spacecraft in Earth?s atmosphere within 100/years. Pareto optimal solutions are obtained using the manoeuvre v and the time-to-reentry as objective functions to be minimised. To explore at the best the search space a semi-analytical orbit propagator, which can propagate an orbit for 100/years in few seconds, is adopted. An in-depth analysis of the results is carried out to understand the conditions leading to a fast reentry with minimum propellant. For this aim a new way of representing the disposal solutions is introduced. With a single 2D plot we are able to fully describe the time evolution of all the relevant orbital parameters as well as identify the conditions that enables the eccentricity build-up. The EoL disposal of the Galileo constellation is used as test case.
Maintaining missions in proximity of small bodies involves extensive orbit determination and ground station time due to the current ground-in-the-loop approach. The prospect of having multiple concurrent missions around different targets requires the development of concepts and capabilities for autonomous proximity operations. Developments in on-board navigation by landmark maps paved the way for autonomous guidance at asteroids. The missing elements for achieving this goal are gravity models, simple enough to be easily used by the spacecraft to steer itself around the asteroid, and guidance laws that rely on such inherently simple models. In this research, we identify a class of models that can represent some characteristics of the dynamical environment around small bodies with sufficient accuracy to enable autonomous guidance. We found that sets of three point masses are suitable to represent the rotational equilibrium points generated by the balance of gravity and centrifugal acceleration in the body-fixed frame. The equilibrium point at the lowest Jacobi energy can be viewed as the energy-gateway to the surface. Information of the location and energy of this point can then be used by a control law to comply with a condition of stability against impact for orbital trajectories. In this thesis, we show an optimisation process for the derivation of three-point mass models from higher order ones and compare the profile of the Zero-Velocity curves between the two models. We define an autonomous guidance law for achieving body fixed hovering in proximity of the asteroid while ensuring that no impact will occur with the small body during the manoeuvre. Finally, we discuss the performance of this approach by comparing it with another autonomous guidance law present in literature and we suggest possible future developments.
Searching for naturally bounded relative orbits in a zonal gravitational field is a crucial
and challenging task in astrodynamics. In this work, a semi-analytical approach based on
high-order Taylor expansions of Poincaré maps is developed. Entire families of periodic orbits,
parameterized by the energy and the polar component of the angular momentum, are computed
under arbitrary order zonal harmonic perturbations, thus enabling the straightforward
design of missions with prescribed properties. The same technique is then proven effective in
determining quasi-periodic orbits that are in bounded relative motion for long time and with
very large aperture. Finally, an illustrative example on how to frame the design of bounded
relative orbits with prescribed properties as an optimization problem is presented.
Ntagiou E, Iacopino C, Policella N, Armellin R, Donati A (2018) Ant-based Mission Planning: Two Examples, Proceedings of 2018 SpaceOps Conference pp. 1-11
American Institute of Aeronautics and Astronautics
The Earth Observation market is growing rapidly, along with the missions' complexity.
Therefore, automated Mission Planning systems are being designed, allowing for operators
to simply specify their intentions on a high level. In this paper, we propose an auto-
mated Mission Planning System based on the ants' foraging mechanism and apply it to
two different mission planning problems, from an Earth Imaging and a Data Relay mission,
investigating the system's ability to be generalised. We compare the planning process for
the two problems and generalise on the type of planning problems the system can address.
A differential algebra based importance sampling method for uncertainty propagation
and impact probability computation on the first resonant returns of Near Earth Objects is presented in this paper. Starting from the results of an orbit determination
process, we use a diferential algebra based automatic domain pruning to estimate
resonances and automatically propagate in time the regions of the initial uncertainty
set that include the resonant return of interest. The result is a list of polynomial state
vectors, each mapping specific regions of the uncertainty set from the observation
epoch to the resonant return. Then, we employ a Monte Carlo importance sampling
technique on the generated subsets for impact probability computation. We assess the
performance of the proposed approach on the case of asteroid (99942) Apophis. A
sensitivity analysis on the main parameters of the technique is carried out, providing
guidelines for their selection. We finally compare the results of the proposed method
to standard and advanced orbital sampling techniques.
Kruzelecky Roman, Murzionak Piotr, Lavoie Jonathan, Sinclair Ian, Schinn Gregory, Underwood Craig, Gao Yang, Bridges Christopher, Armellin Roberto, Luccafabris Andrea, Cloutis Edward, Leijtens Johan (2018) VMMO Lunar Volatile and Mineralogy Mapping Orbiter, International Conference on Environmental Systems Proceedings 2018
MPB Communications Inc.
Understanding the lunar near-surface distribution of relevant in-situ resources, such as ilmenite (FeTiO3), and volatiles, such as water/ice, is vital to future sustained manned bases. VMMO is a highly-capable, low-cost 12U Cubesat designed for operation in a lunar frozen orbit. It accomodates the LVMM Lunar Volatile and Mineralogy Mapper and the CLAIRE Compact LunAr Ionising Radiation Environment payloads. LVMM is a multi-wavelength Chemical Lidar using fiber lasers emitting at 532nm and 1560nm, with an optional 1064nm channel, for stand-off mapping of the lunar ice distribution using active laser illumination, with a focus on the permanently-shadowed craters in the lunar south pole. This combination of spectral channels can provide sensitive discrimination of water/ice in various regolith. The fiber-laser technology has heritage in the ongoing Fiber Sensor Demonstrator flying on ESA's Proba-2. LVMM can also be used in a low-power passive mode with an added 280nm UV channel to map the lunar mineralogy and ilmenite distribution during the lunar day using the reflected solar illumination. CLAIRE is designed to provide a highly miniaturized radiation environment and effect monitor. CLAIRE draws on heritage from the MuREM and RM payloads, flown on the UK?s TDS-1 spacecraft. The payload includes PIN-diode sensors to measure ionizing particle fluxes (protons and heavy-ions) and to record the resulting linear energy transfer (LET) energy-deposition spectra. It also includes solid-state RADFET dosimeters to measure accumulated ionizing dose, and dose-rate diode detectors, designed to respond to a Coronal Mass Ejection (CME) or Solar Particle Event (SPE). CLAIRE also includes an electronic component test board, capable of measuring SEEs and TID effects in a selected set of candidate electronics, allowing direct correlations between effects and the real measured environment.
Ntagiou Evridiki V., Iacopino Claudio, Policella Nicola, Armellin Roberto, Donati Alessandro (2017) Coverage Planning for Earth Observation Constellations, Proceedings of the 11th, Scheduling and Planning Applications, woRKshop (SPARK), June 19-23, 2017, Pittsburgh, USA pp. 41-48
Earth Observation market has been increasing in both size
and complexity over the last years. EO missions are
becoming more capable and more agile, carrying highresolution
sensors that need to frequently be steered at
different directions depending on the mission goals. In this
paper we discuss the Coverage Planning problem Disaster
Monitoring Constellation (DMC3) mission deals with. It is
an Earth Imaging mission from Surrey Satellite Technology
Ltd (SSTL). The combinatorial optimization problem of
determining a not only feasible but optimal sequence of the
spacecraft attitude in order to image the total of a target area
is NP-hard. We propose an automated planning system for
DMC3, employing a self-organizing software architecture
and a nature inspired optimization algorithm, Ant Colony
Optimization. The advantages of the system are discussed
and some key results are shown.
Pirovano Laura, Santeramo Daniele Antonio, Armellin Roberto, di Lizia Pierluigi, Wittig Alexander (2018) Probabilistic data association based on intersection of Orbit Sets, Proceedings of the 19th AMOS Conference, September 11-14, 2018, Maui, USA
Maui Economic Development Board, Inc.
Untraced space debris are the principal threat to the functioning of operational satellites whose services have become
a fundamental part of our daily life. Small debris between 1 and 10 cm are currently too small to be cataloged and are
only detectable for a limited amount of time when surveying the sky. The very-short arc nature of the observations
makes it very difficult to perform precise orbit determination with only one passage of the object over the observing
station. For this reason the problem of data association becomes relevant: one has to find more observations of the same
resident space object to precisely determine its orbit. This paper is going to illustrate a novel approach that exploits
Differential Algebra to handle the data association problem in a completely analytical way. The paper presents an
algorithm that uses the Subset Simulation to find correlated observations starting from the solution to the Initial Orbit
Determination problem. Due to the different capabilities of the observatories, several observing strategies are currently
being used. The algorithm is thus tested for different strategies against a well known approach from literature. Then,
the performance of the data association method is tested on some real observations obtained in two consecutive nights.
Finally, preliminary results for data association without Initial Orbit Determination are shown
One of the objectives of mission design is selecting an optimum orbital transfer which often translated as a transfer which requires minimum propellant consumption. In order to assure the selected trajectory meets the requirement, the optimality of transfer should first be analyzed either by directly calculating the ?V of the candidate trajectories and select the one that gives a minimum value or by evaluating the trajectory according to certain criteria of optimality. The second method is performed by analyzing the profile of the modulus of the thrust direction vector which is known as primer vector. Both methods come with their own advantages and disadvantages. However, it is possible to use the primer vector method to verify if the result from the direct method is truly optimal or if the ?V can be reduced further by implementing correction maneuver to the reference trajectory. In addition to its capability to evaluate the transfer optimality without the need to calculate the transfer ?V, primer vector also enables us to identify the time and position to apply correction maneuver in order to optimize a non-optimum transfer. This paper will present the analytical approach to the fuel optimal impulsive transfer using primer vector method. The validity of the method is confirmed by comparing the result to those from the numerical method. The investigation of the optimality of direct transfer is used to give an example of the application of the method. The case under study is the prograde elliptic transfers from Earth to Mars. The study enables us to identify the optimality of all the possible transfers.
The propagation and Poincaré mapping of perturbed Keplerian
motion is a key topic in celestial mechanics and astrodynamics, e.g. to study
the stability of orbits or design bounded relative trajectories. The high-order
transfer map (HOTM) method enables efficient mapping of perturbed Kep-
lerian orbits using the high-order Taylor expansion of a Poincaré or strobo-scopic map. The HOTM is only accurate close to the expansion point and
therefore the number of revolutions for which the map is accurate tends to
be limited. The proper selection of coordinates is of key importance for im-
proving the performance of the HOTM method. In this paper, we investigate
the use of different element sets for expressing the high-order map in order
to find the coordinates that perform best in terms of accuracy. A new set of
elements is introduced that enables extremely accurate mapping of the state,
even for high eccentricities and higher-order zonal perturbations. Finally, the
high-order map is shown to be very useful for the determination and study of fixed points and center manifolds of Poincaré maps.
Optimal orbital trajectories are obtained through the solution of highly nonlinear large-scale problems. In the case of low-thrust propulsion applications, the spacecraft benefits from high specific impulses and, hence, greater payload mass. However, these missions require a high count of orbital revolutions and, therefore, display augmented sensitivity to many disturbances. Solutions to such problems can be tackled via a discrete approach, using optimal feedback control laws. Historically, differential dynamic programming (DDP) has shown outstanding results in tackling these problems. A state of the art software that implements a variation of DDP has been developed by Whiffen (2006) and it is used by NASA?s DAWN mission. One of the latest techniques implemented to deal with these discrete constrained optimizations is the Hybrid Differential Dynamic Programming (HDDP) algorithm, introduced by Lantoine and Russell (2012). This method complements the reliability and efficiency of classic nonlinear programming techniques with the robustness to poor initial guesses and the reduced computational effort of DDP. The key feature of the algorithm is the exploitation of a second order state transition matrix procedure to propagate the needed partials, decoupling the dynamics from the optimization. In doing so, it renders the integration of dynamical equations suitable for parallelization. Together with the possibility to treat constrained problems, this represents the greatest improvement of classic DDP. Nevertheless, the major limitation of this approach is the high computational cost to evaluate the required state transition matrices. Analytical derivatives, when available, have shown a significant reduction in the computational cost and time for HDDP application. This work applies differential algebra (DA) to HDDP to cope with this limitation. DA is introduced to obtain state transition matrices as polynomial maps. These maps come directly from the integration of the dynamics of the system, removing the dedicated algorithmic step and reducing its computational cost. Moreover, by operating on polynomial maps, all the solutions of local optimization problems are treated through differential algebraic techniques. This approach allows users to deal with higher order expansions of the cost, without modifying the algorithm. From the examples provided, it emerges that increasing the order of the expansions does not yield a better convergence rate. Additionally, it causes numerical instability of the algorithm to arise, as well as a noticeable increase on computational time due to the number of polynomial coefficients that ought to be computed.
Uncertainty propagation has been addressed extensively in space missions around Earth. For small solar system missions, new challenges come forth mainly due to the generally irregular and weak gravity field and larger uncertainties in system parameters, e.g. gravity field, density and rotation rate etc. Nevertheless, the study on state propagation in the uncertain gravity field of small body is limited and is the focus of this study. To both address the efficiency and meet the required accuracy, this study applies the dynamics-based differential algebra (DA) method to propagate the state of the spacecraft (s/c) in the uncertain gravity field of the small body. The DA allows arbitrary order Taylor expansion of the flow of the highly nonlinear dynamics w.r.t. the uncertain parameters of the dynamics, and fast evaluations of the flow. With the Gaussian distributions of the initial uncertainties, the distribution of the final state are obtained by transforming through the selected order Taylor expansion w.r.t. the second order gravity. Taking asteroid Stein as an example, the error due to the uncertain second order gravity are analysed for different orbital geometry. The retrograde motion is more robust to the uncertainty than the prograde case. The discoveries of this research can help mission designers assessing the posed risks and designing appropriate mission strategies.
Predicting re-entry epoch of space objects enables managing the risk to
ground population. Predictions are particularly difficult for objects in highlyelliptical
orbits, and important for objects with components that can survive
re-entry, e.g. rocket bodies (R/Bs). This paper presents a methodology to
filter two-line element sets (TLEs) to facilitate accurate re-entry prediction
of such objects. Difficulties in using TLEs for precise analyses are highlighted
and a set of filters that identifies erroneous element sets is developed. The
filter settings are optimised using an artificially generated TLE time series.
Optimisation results are verified on real TLEs by analysing the automatically
found outliers for exemplar R/Bs. Based on a study of 96 historical
re-entries, it is shown that TLE filtering is necessary on all orbital elements
that are being used in a given analysis in order to avoid considerably inaccurate
The continuation of space activities is at risk due to the growing number of uncontrolled objects, called space debris, which can collide with operational spacecraft. In addition, debris can fall back to the Earth causing risks to the population. Therefore, space agencies have started space situational awareness (SSA) programs and taken space debris mitigation measures to reduce the risks caused by uncontrolled objects and prevent the generation of new debris. A fundamental need for SSA is the capability to predict, design and analyse orbits. In this work, new techniques for orbit prediction are developed that are suitable for SSA in terms of accuracy, efficiency and ability to deal with uncertainties and are applied for re-entry prediction, end-of-life disposal, ADR mission design and long-term orbit prediction. The performance of high-order Poincaré mapping of perturbed orbits is improved by introducing a new set of orbital elements and the method is applied for orbit propagation and analysis of quasi-periodic orbits. Two new Lambert problem solvers are developed to compute perturbed rendezvous trajectories with hundreds of revolutions for the design of active debris removal missions. The computation of the effect of drag for semi-analytical propagation is speed up by using high-order Taylor expansions to evaluate the mean element rates efficiently. In addition, the high-order expansion of the flow through semi-analytical propagation is enabled using differential algebra to allow efficient propagation of initial conditions. The predictability of Galileo disposal orbits was investigated using chaos indicators and sensitivity analysis. The study showed that the orbits are predictable and that chaos indicators are not unsuitable for predictability analysis. Finally, to improve the re-entry prediction of rocket bodies based on two-line element data, ballistic coefficient and state estimation methods are enhanced. Using the developed approach, the re-entry prediction using only a ballistic coefficient estimate was found to be as accurate as re-entry prediction after full state estimation.
Untraced space debris are the principal threat to the functioning
of operational satellites whose services have become
a fundamental part of our daily life. Small debris
between 1 and 10 cm are currently too small to be cataloged
and are only detectable for a limited amount of
time when surveying the sky. The very-short arc nature
of the observations makes it very difficult to perform precise
orbit determination with only one passage of the object
over the observing station. For this reason the problem
of data association becomes relevant: one has to find
more observations of the same resident space object to
precisely determine its orbit. This paper focuses on multitarget
tracking, which is part of the data association problem
and deals with the challenge of jointly estimating the
number of observed targets and their states from sensor
data. We propose a new method that builds on the admissible
region approach and exploits differential algebra
to efficiently estimate uncertainty ranges to discriminate
between correlated and uncorrelated observations. The
multi-target tracking problem is formulated with two different
mathematical conditions: as initial-value problem
and as boundary-value problem. The first one allows us
to define the constraints as a six-dimensional region at a
single epoch for each observation, while the second one,
instead, allows us to consider the two-by-two comparison
as a Lamberts problem thus constraining the position
vectors at the two epochs. The efficiency and success rate
of the two formulations is then evaluated.
A semi-analytical technique for both the design and control of repeat groundtrack
(RGT) orbits in a high fidelity dynamical model, including non-conservative
forces and accurate Earth orientation parameters, is introduced. The method is
based on the use of high-order expansion of Poincaré maps to propagate forward in
time regions of the phase space for one, or more, repeat cycles. This map provides
the means to efficiently study the effect that an impulse, applied at the Poincaré
section crossing, produces on the ground-track pattern, thus enabling highly accurate
design and control. The approach is applied to the design and control of
missions like TerraSAR-X, Landsat-8, SPOT-7, IRS-P6, and UoSAT-12.
The Earth Observation (EO) market is rapidly growing due to technology miniaturization,
cheaper launch opportunities and wider spectrum of EO applications. Along with an exponential
growth of the ground based users that can access Low Earth Orbit (LEO) spacecraft
data, this growing community represents an important demand for Data Relay missions. LEO
spacecraft have short visibility windows to the Ground Stations (GS) which limit their throughput.
Data Relay missions are comprised of spacecraft at higher altitude orbits (Geostationary
Orbits) acting as relays of data among LEO spacecraft and GS. Those missions are then able
to offer more frequent data downlink opportunities to the LEO spacecraft thus increasing
the volume of the data reaching the ground and improving the responsiveness between users?
requests and downlink operations. Ground based Mission Planning Systems (MPS) are commonly
managing such complex missions, representing a large operational cost. In this paper,
we propose the application of a Swarm Intelligence algorithm to the design of an automated
MPS for Data Relay missions. Automated MPS have the potential of saving operational costs
while leaving the high level decisions to human operators. This paper represents the first time
that an Ant Colony Optimization (ACO) algorithm is applied to this type of scheduling problem.
This family of algorithms is generally found to offer good level of efficiency and scalability. In
this work, we compare our approach against an algorithm popular in literature, called Squeaky
Wheel Optimization (SWO) and show how our ACO algorithm outperforms it.
Currently, the explorations of small solar system bodies (asteroids and comets) have become more and more popular. Due to the limited measurement capability and irregular shape and diverse spin status of the small body, uncertainties on the parameters of the system and s/c executions are a practical and troublesome problem for mission design and operations. The sample-based Monte Carlo simulation is primarily used to propagate and analyze the effects of these uncertainties on the surrounding orbital motion. However, it is generally time-consuming because of large samples required by the highly nonlinear dynamics. New methods need to be applied for balancing computational efficiency and accuracy. To motivate this research area and facilitate the mission design process, this review firstly discusses the dynamical models and the different methods of modeling the mostly related gravitational and non-gravitational forces. Then the main uncertainties in these force models are classified and analyzed, including approaching, orbiting and landing. Then the linear and nonlinear uncertainty propagation methods are described, together with their advantages and drawbacks. Typical mission examples and the associated uncertainty analysis, in terms of methods and outcomes, are summarized. Future research efforts are emphasized in terms of complete modelling, new mission scenarios, and application of (semi-) analytical methods in small body explorations.
Uncertainty propagation has been addressed extensively in space missions around the Earth, but much less for missions around small solar system bodies. Small bodies usually have irregular and weak gravity and our knowledge of their gravity, rotation speed and surrounding space environment is largely uncertain. These characteristics make the orbit propagation around these small bodies a challenging task.
Focusing on the uncertainty of the small body's gravity, this paper applies the differential algebra (DA) technique to study the orbit propagation problem, and addresses its efficiency for a given the required accuracy. Different from traditional studies that focus on the uncertainty of the initial state, this study assumes an exact initial state and studies the influences that gravity model uncertainties have on the orbit. Taking the asteroid Steins as an example, the accuracy and the efficiency of the DA approach are firstly validated by comparison with the traditional Monte Carlo method. Then, the effects of gravity uncertainties on different types of orbits (prograde, retrograde and polar) are studied. The retrograde motion is found to be more robust to the gravity uncertainty than the prograde ones. For near polar orbits, the impact of gravity uncertainty on orbital motion depends significantly on the initial position, and it reaches the maximum if the initial position is near the polar regions. Moreover, short-term effects are found to play an important role in orbit deviation as a result of the gravity uncertainty. These discoveries can help mission designers assess the posed risk and design appropriate mission orbits.
Missions involving multiple spacecraft have been in the spotlight during the last years due to the number of scientific and engineering advantages they offer. They are crucial in the context of Earth monitoring, disaster management and data relay missions which provide
a way of addressing the increasing demand of the volume of data produced on-board daily. The spacecraft are equipped with complex instruments of many capabilities and
diverse constraints which exponentially increase the possible state of the constellation at each moment. Hence, their management has become too complex for human operators to handle. To this end, automated mission planning systems are designed and implemented; the operator defines a goal on a higher level, i.e. submits an imaging request, and the
system is responsible for determining the activities to achieve it, taking into account the mission's capabilities and constraints. The main challenge when designing such systems lies in finding methods that reliably produce optimal or near optimal solutions while also satisfying certain scalability and responsiveness system requirements.
In this work, we consider two different target missions and propose a ground-based space mission planning framework to address their planning problems. A coverage planning problem for the Disaster Monitoring Constellation 3 mission from Surrey Satellite Technology Limited is solved; an Earth imaging mission consisting of 3 agile spacecraft, that is expected to image 1 million square km each day. We also address the planning problem of a data relay mission, where Geostationary spacecraft act as relays of data among low Earth orbit spacecraft and ground stations. The
flexible LEO spacecraft requests have different priorities and are far more than the GEO spacecraft can accommodate, resulting in an oversubscribed scheduling problem. Both problems are of very high computational complexity for exact methods to be efficient. We explore the potential of a stochastic algorithm that is based on the ants' foraging mechanism, Ant Colony Optimisation. Applying this nature-inspired technique to the two planning problems, we manage to achieve optimisation of their schedules while coordinating the constellation's spacecraft. In order to further investigate the capabilities of the method,
we introduce and analyse the stability of a non-linear system that models the long-term ACO dynamics in directed graph environments. The analysis provided insights that have proved helpful in increasing the algorithm's efficiency. The performance of the designed system is validated by comparing our results with a Squeaky Wheel Optimisation approach, a method widely used in recent years in the space mission planning field.
Untraced space debris is the principal threat to the functioning of operational satellites whose services have become
a fundamental part of our daily life. Small debris between 1 and 10 cm are currently too small to be cataloged and
are only detectable for a limited amount of time. Indeed, when performing survey campaigns, the selected schedule
and/or visibility constraints often result in short-arc observations with long observing gaps, which do not allow for
precise orbit determination during a single passage of the object over an observing station. For this reason the problem
of data association becomes relevant: one has to find more observations of the same resident space object to precisely
determine its orbit. This paper compares different methods that build on the concept of Admissible Region and
attributable to solve the problem of correlating sparse observations. The comparison is carried out on real optical
observations obtained by the ZimSMART telescope on consecutive nights.
Low-thrust many-revolution trajectory design and orbit transfers are becoming increasingly
important with the development of high specific impulse, low-thrust engines. Closed-loop
feedback-driven (CLFD) control laws can be used to solve these trajectory design problems
with minimal computational cost and offer potential for autonomous guidance. However,
they have user-defined parameters which limit their optimality. In this work, an actor-critic
reinforcement learning framework is proposed to make the parameters of the Lyapunov-based
Q-law state-dependent, ensuring the controller can adapt as the dynamics evolve during a
transfer. The proposed framework should be independent of the particular CLFD control
law and provides improved solutions for mission analysis. There is also potential for future
on-board autonomous use, as trajectories are closed-form and can be generated without an
initial guess. The current results focus on GTO-GEO transfers in Keplerian dynamics and
later with eclipse and J2 effects. Both time-optimal and mass-optimal transfers are presented,
and the stability to uncertainties in orbit determination are discussed. The task of handling
orbit perturbations is left to future work.
Space activities are continuously at risk due to the ever-increasing amount of uncontrolled space objects, called space debris, which can collide with operational spacecraft. Therefore, space agencies have started space situational awareness (SSA) programs, which aim to provide timely and accurate data regarding the space environment and hazards to infrastructure in orbit and on the ground. An essential need for SSA is the orbit determination of as many space objects as possible, in order to predict and prevent collisions. In this work, techniques for orbit determination of objects observed on short arcs are developed. This scenario frequently occurs with space debris, which are characterised by long observational gaps due to observability constraints. The classical least squares method, used for orbit determination, is revisited by studying the effect of nonlinearities in the mapping between observations and state. Differential algebra techniques are exploited, which enable the computation of high-order Taylor expansions of the residuals with respect to the state. Approximations of differential correction methods can then be avoided and the confidence region of the solution accurately characterized with a nonlinear approach. This uncertainty region can be reduced when more observations of the same objects are acquired. Thus, a method that both recognizes associated observations and sequentially reduces the solution uncertainty when two or more sets of observations are associated is proposed. The six dimensional ($6D$) association problem is addressed as a series of $2D$ and $4D$ problems to alleviate the computational cost. In case of successful association, part of the initial uncertainty is pruned away. The least squares solution and its uncertainty region can finally be used to initialize a sequential filter. A particle filter is implemented using differential algebra, so that system and measurement equations are substituted by polynomial evaluations and the resulting computational cost of the algorithm is alleviated. A population-based stochastic optimizer, the multiobjective particle swarm optimizer, is exploited to generate particles that best represent the probability distribution over solution orbits.