87 research outputs found
Long-term orbital propagation through differential algebra transfer maps and averaging semi-analytical approaches
A differential algebra based importance sampling method for impact probability computation on Earth resonant returns of Near Earth Objects
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 differential 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
Collision avoidance maneuver design based on multi-objective optimization
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 Δv, 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 Δv 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
Dealing with uncertainties in angles-only initial orbit determination
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
Nonlinear Uncertainty Propagation in Astrodynamics Using Differential Algebra and Graphics Processing Units
In this paper, two numerical methods for nonlinear uncertainty propagation in astrodynamics are presented and thoroughly compared. Both methods are based on the Monte Carlo idea of evaluating multiple samples of an initial statistical distribution around the nominal state. However, whereas the graphics processing unit implementation aims at increasing the performances of the classical Monte Carlo approach exploiting the massively parallel architecture of modern general-purpose computing on graphics processing units, the method based on differential algebra is aimed at the improvement and generalization of standard linear methods for uncertainty propagation. The two proposed numerical methods are applied to test cases considering both simple two-body dynamics and a full n-body dynamics with accurate ephemeris. The results of the propagation are thoroughly compared with particular emphasis on both accuracy and computational performances
Analytical framework for space debris collision avoidance maneuver design
An analytical formulation for collision avoidance maneuvers involving a
spacecraft and a space debris is presented, including solutions for the maximum
deviation and minimum collision probability cases. Gauss' planetary equations
and relative motion equations are used to map maneuvers at a given time to
displacements at the predicted close approach. The model is then extended to
map changes in state between two times, allowing one to propagate covariance
matrices. The analytical formulation reduces the optimization problem to an
eigenproblem, both for maximum deviation and minimum collision probability. Two
maximum deviation cases, total deviation and impact parameter, are compared for
a large set of spacecraft-debris conjunction geometries derived from European
Space Agency's Meteoroid and Space Debris Terrestrial Environment Reference
(MASTER-2009) model. Moreover, the maximum impact parameter and minimum
collision probability maneuvers are compared assuming covariances known at the
maneuver time, to evaluate the net effect of lead time in collision
probability. In all cases, solutions are analyzed in the b-plane to leverage
its natural separation of phasing and geometry change effects. Both
uncertainties and maximum deviation grow along the time axis for long lead
times, limiting the reduction in collision probability.Comment: 44 pages, 24 figures. Author's accepted manuscript for a paper
published in Journal of Guidance, Control, and Dynamic
Impact probability computation of Near-Earth Objects using Monte Carlo Line Sampling and Subset Simulation
This work introduces two Monte Carlo (MC)-based sampling methods, known as
line sampling and subset simulation, to improve the performance of standard MC
analyses in the context of asteroid impact risk assessment. Both techniques
sample the initial uncertainty region in different ways, with the result of
either providing a more accurate estimate of the impact probability or reducing
the number of required samples during the simulation with respect to standard
MC techniques. The two methods are first described and then applied to some
test cases, providing evidence of the increased accuracy or the reduced
computational burden with respect to a standard MC simulation. Finally, a
sensitivity analysis is carried out to show how parameter setting affects the
accuracy of the results and the numerical efficiency of the two methods
Long term nonlinear propagation of uncertainties in perturbed geocentric dynamics using automatic domain splitting
Current approaches to uncertainty propagation in astrodynamics mainly refer tolinearized models or Monte Carlo simulations. Naive linear methods fail in nonlinear dynamics, whereas Monte Carlo simulations tend to be computationallyintensive. Differential algebra has already proven to be an efficient compromiseby replacing thousands of pointwise integrations of Monte Carlo runs with thefast 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
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