165 research outputs found
Element sets for high-order Poincar\'e mapping of perturbed Keplerian motion
The propagation and Poincar\'e 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 Keplerian orbits over many
revolutions. For this, the method uses the high-order Taylor expansion of a
Poincar\'e or stroboscopic map, which is accurate close to the expansion point.
In this paper, we investigate the performance of the HOTM method using
different element sets for building the high-order map. The element sets
investigated are the classical orbital elements, modified equinoctial elements,
Hill variables, cylindrical coordinates and Deprit's ideal elements. The
performances of the different coordinate sets are tested by comparing the
accuracy and efficiency of mapping low-Earth and highly-elliptical orbits
perturbed by with numerical propagation. The accuracy of HOTM depends
strongly on the choice of elements and type of orbit. 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 centre manifolds of Poincar\'e maps.Comment: Pre-print of journal articl
Semi-analytical guidance algorithm for autonomous close approach to non-cooperative low-gravity targets
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
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
Survey on studies about model uncertainties in small body explorations
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
A low-order automatic domain splitting approach for nonlinear uncertainty mapping
This paper introduces a novel method for the automatic detection and handling
of nonlinearities in a generic transformation. A nonlinearity index that
exploits second order Taylor expansions and polynomial bounding techniques is
first introduced to rigorously estimate the Jacobian variation of a nonlinear
transformation. This index is then embedded into a low-order automatic domain
splitting algorithm that accurately describes the mapping of an initial
uncertainty set through a generic nonlinear transformation by splitting the
domain whenever some imposed linearity constraints are non met. The algorithm
is illustrated in the critical case of orbital uncertainty propagation, and it
is coupled with a tailored merging algorithm that limits the growth of the
domains in time by recombining them when nonlinearities decrease. The low-order
automatic domain splitting algorithm is then combined with Gaussian mixtures
models to accurately describe the propagation of a probability density
function. A detailed analysis of the proposed method is presented, and the
impact of the different available degrees of freedom on the accuracy and
performance of the method is studied
A Koopman-Operator Control Optimization for Relative Motion in Space
A high order optimal control strategy implemented in the Koopman operator
framework is proposed in this work. The new technique exploits the Koopman
representation of the solution of the equations of motion to develop an energy
optimal inverse control methodology. The operator theory can reformulate a
nonlinear dynamical system of finite dimension into a linear system with an
infinite number of dimensions. As a results, the state of any nonlinear
dynamics is represented as a linear combination of high-order orthogonal
polynomials, which creates the state transition polynomial map of the solution.
Since the optimal control technique can be reduced to a two-points boundary
value problem, the Koopman map is used to connect the state and control
variables in time, such that optimal values are obtained through map inversion
and polynomial evaluation. The new technique is applied to rendezvous
applications in space, where the relative motion between two satellites is
modelled with a high-order polynomial series expansion of the Lagrangian of the
system, such that the Clohessy-Wiltshire equations represent the reduction of
the high-order model to a linear truncation.Comment: 19 pages, 9 figures, 2023 SciTech conference. arXiv admin note: text
overlap with arXiv:2111.0748
Confidence region of least squares solution for single-arc observations
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 approximationsof 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
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