共查询到20条相似文献，搜索用时 31 毫秒
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Using alternative independent variables in lieu of time has important advantages when propagating the partial derivatives of the trajectory. This paper focuses on spacecraft relative motion, but the concepts presented here can be extended to any problem involving the variational equations of orbital motion. A usual approach for modeling the relative dynamics is to evaluate how the reference orbit changes when modifying the initial conditions slightly. But when the time is a mere dependent variable, changes in the initial conditions will result in changes in time as well: a time delay between the reference and the neighbor solution will appear. The theory of asynchronous relative motion shows how the time delay can be corrected to recover the physical sense of the solution and, more importantly, how this correction can be used to improve significantly the accuracy of the linear solutions to relative motion found in the literature. As an example, an improved version of the Clohessy-Wiltshire (CW) solution is presented explicitly. The correcting terms are extremely compact, and the solution proves more accurate than the second and even third order CW equations for long propagations. The application to the elliptic case is also discussed. The theory is not restricted to Keplerian orbits, as it holds under any perturbation. To prove this statement, two examples of realistic trajectories are presented: a pair of spacecraft orbiting the Earth and perturbed by a realistic force model; and two probes describing a quasi-periodic orbit in the Jupiter-Europa system subject to third-body perturbations. The numerical examples show that the new theory yields reductions in the propagation error of several orders of magnitude, both in position and velocity, when compared to the linear approach. 相似文献
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M. Sabatini D. Izzo G. B. Palmerini 《Celestial Mechanics and Dynamical Astronomy》2009,105(1-3):141-157
Large ΔV amounts are often required to maintain the relative geometry which is needed to implement a formation flying concept. A wise use of the orbital environment makes the orbit keeping phase easier, reducing the overall propellant consumption. A first important step in this direction is the selection of formation configurations and orbits which, while still satisfying the mission requirements, are less subject to perturbations resulting naturally in closed relative motion. Within this frame, a number of studies have been recently carried out in order to identify possible sets of invariant relative orbits under the effects of the Earth oblateness, a conservative force commonly referred to as J2 which is also the most important perturbation for Low Earth Orbit. These efforts clearly marked the difficulties connected with achieving genuine periodic relative motion under J2 effect, but they also showed the existence of a set of conditions on the orbital parameters which allow for quasi-periodic J2 trajectories. This paper presents these particular trajectories, by means of deeper theoretical explanations, showing the dependency of the shape of the relative configurations on the orbital inclination. Since the quasi-periodic trajectories cannot be written analytically, and moreover, they are very sensitive with respect to the initial conditions, difficulties arise when trying to exploit these paths as reference for the control of a formation. This paper proposes a novel approach to find, from the actual quasi periodic natural trajectories, minimal control periodic reference trajectories. Next, it evaluates quantitatively the amount of propellant which is needed to control a spacecraft formation along these paths. The choice of Hill’s classical circular projected configuration as a nominal trajectory is considered as a comparison, showing the clear advantages of the proposed guidance design, which assumes low-perturbed periodic reference orbits as nominal trajectories. 相似文献
4.
In this paper we present a new method of restoration of the true three-dimensional (3D) trajectories of prominence knots using
ground-based observations taken with a single telescope that is equipped with a Multi-Channel Subtractive Double Pass imaging
spectrograph. Our method allows us to evaluate the true 3D trajectories of the prominence knots without any assumptions concerning
the shape of the trajectories or the dynamics of the motion. The reconstructed trajectories of several knots observed in three
prominences are presented. 相似文献
5.
K. Uldall Kristiansen P. L. Palmer M. Roberts 《Celestial Mechanics and Dynamical Astronomy》2010,106(4):371-390
This paper builds upon the work of Palmer and Imre exploring the relative motion of satellites on neighbouring Keplerian orbits.
We make use of a general geometrical setting from Hamiltonian systems theory to obtain analytical solutions of the variational
Kepler equations in an Earth centred inertial coordinate frame in terms of the relevant conserved quantities: relative energy,
relative angular momentum and the relative eccentricity vector. The paper extends the work on relative satellite motion by
providing solutions about any elliptic, parabolic or hyperbolic reference trajectory, including the zero angular momentum
case. The geometrical framework assists the design of complex formation flying trajectories. This is demonstrated by the construction
of a tetrahedral formation, described through the relevant conserved quantities, for which the satellites are on highly eccentric
orbits around the Sun to visit the Kuiper belt. 相似文献
6.
The two-body problem is a twelfth-order time-invariant dynamic system, and therefore has eleven mutually-independent time-independent
integrals, here referred to as motion constants. Some of these motion constants are related to the ten mutually-independent
algebraic integrals of the n-body problem, whereas some are particular to the two-body problem. The problem can be decomposed into mass-center and relative-motion
subsystems, each being sixth-order and each having five mutually-independent motion constants. This paper presents solutions
for the eleventh motion constant, which relates the behavior of the two subsystems. The complete set of mutually-independent
motion constants describes the shape of the state-space trajectories. The use of the eleventh motion constant is demonstrated
in computing a solution to a two-point boundary-value problem. 相似文献
7.
This paper studies the relative motion of satellite formation flying in arbitrary elliptical orbits with no perturbation.
The trajectories of the leader and follower satellites are projected onto the celestial sphere. These two projections and
celestial equator intersect each other to form a spherical triangle, in which the vertex angles and arc-distances are used
to describe the relative motion equations. This method is entitled the reference orbital element approach. Here the dimensionless
distance is defined as the ratio of the maximal distance between the leader and follower satellites to the semi-major axis
of the leader satellite. In close formations, this dimensionless distance, as well as some vertex angles and arc-distances
of this spherical triangle, and the orbital element differences are small quantities. A series of order-of-magnitude analyses
about these quantities are conducted. Consequently, the relative motion equations are approximated by expansions truncated
to the second order, i.e. square of the dimensionless distance. In order to study the problem of periodicity of relative motion,
the semi-major axis of the follower is expanded as Taylor series around that of the leader, by regarding relative position
and velocity as small quantities. Using this expansion, it is proved that the periodicity condition derived from Lawden’s
equations is equivalent to the condition that the Taylor series of order one is zero. The first-order relative motion equations,
simplified from the second-order ones, possess the same forms as the periodic solutions of Lawden’s equations. It is presented
that the latter are further first-order approximations to the former; and moreover, compared with the latter more suitable
to research spacecraft rendezvous and docking, the former are more suitable to research relative orbit configurations. The
first-order relative motion equations are expanded as trigonometric series with eccentric anomaly as the angle variable. Except
the terms of order one, the trigonometric series’ amplitudes are geometric series, and corresponding phases are constant both
in the radial and in-track directions. When the trajectory of the in-plane relative motion is similar to an ellipse, a method
to seek this ellipse is presented. The advantage of this method is shown by an example. 相似文献
8.
Vadim A. Antonov Faziliddin T. Shamshiev 《Celestial Mechanics and Dynamical Astronomy》1994,59(3):209-219
The paper deals with some special cases of the existence of a local integral of motion in a two-dimensional potential field rotating with constant angular velocity. In this case the trajectories may be completely determined, which is not always possible in other cases with a local integral, in contrast to the cases with a true integral. Some cases where the trajectories can be determined analytically are trivial but there are also some new nontrivial cases. 相似文献
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Zhen Yang Yazhong Luo Jin Zhang Hengnian Li 《Celestial Mechanics and Dynamical Astronomy》2018,130(9):56
This paper develops a nonlinear analytic solution for satellite relative motion in J2-perturbed elliptic orbits by using the geometric method that can avoid directly solving the complex differential equations. The differential equinoctial elements (DEEs) are used to remove any singularities for zero-eccentricity or zero-inclination orbits. Based on the relationship between the relative states and the DEEs, state transition tensors (STTs) for transforming the osculating DEEs and propagating the mean DEEs have been derived. The formulation of these STTs has been split into a set of vector and matrix operations, which avoids directly expanding the complex second-order terms, and thus, the obtained STTs could be easy-to-understand and easy-to-code. Numerical results show that the proposed nonlinear solution is valid for zero-eccentricity and zero-inclination reference orbit and is more accurate than the previous linear or nonlinear methods for the long-term prediction of satellite relative motion. 相似文献
11.
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 Keplerian orbits using the high-order Taylor expansion of a Poincaré or stroboscopic 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 improving 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. 相似文献
12.
The problem of finding natural bounded relative trajectories between the different units of a distributed space system is of great interest to the astrodynamics community. This is because most popular initialization methods still fail to establish long-term bounded relative motion when gravitational perturbations are involved. Recent numerical searches based on dynamical systems theory and ergodic maps have demonstrated that bounded relative trajectories not only exist but may extend up to hundreds of kilometers, i.e., well beyond the reach of currently available techniques. To remedy this, we introduce a novel approach that relies on neither linearized equations nor mean-to-osculating orbit element mappings. The proposed algorithm applies to rotationally symmetric bodies and is based on a numerical method for computing quasi-periodic invariant tori via stroboscopic maps, including extra constraints to fix the average of the nodal period and RAAN drift between two consecutive equatorial plane crossings of the quasi-periodic solutions. In this way, bounded relative trajectories of arbitrary size can be found with great accuracy as long as these are allowed by the natural dynamics and the physical constraints of the system (e.g., the surface of the gravitational attractor). This holds under any number of zonal harmonics perturbations and for arbitrary time intervals as demonstrated by numerical simulations about an Earth-like planet and the highly oblate primary of the binary asteroid (66391) 1999 KW4. 相似文献
13.
A detailed study of the motion of test-particles [either having a nonzero rest-mass or zero rest-mass] has been carried out for Tolman's type VII solution with vanishing surface density, which is one of the few physically relevant exact solutions of Einstein's field equations for static and spherically symmetric mass distributions.The trapping angles, 0, at which the test-particles escape these configurations are calculated and the trajectories of test-particles are studied in detail. The types of trajectories found for this spacetime geometry are: (1) arc-like trajectories, (2) spiral-like trajectories ending into a circle of constant radius, (3) trajectories with a cusp at the minimum distance, (4) double-bounded trajectories, and (5) double-bounded trajectories with a cusp at the minimum distance. The test-particles following the trajectories of type (1) and type (3) escape the configuration while others are trapped within the structure. Besides the advancement of periastron of the orbits the retrogradation of periastrons are also observed.The stability of the structures is considered by using the variational method (Chandrasekhar, 1964a,b), and its is seen that the structures remain stable at least for a central redshift,z0, as large as 5.09.These studies may distinguish relativistic cluster from a Newtonian one and may find application to the local models of quasi-stellar objects (QSOs). 相似文献
14.
Ming Xu Jiamin Zhu Tian Tan Shijie Xu 《Celestial Mechanics and Dynamical Astronomy》2012,113(4):403-433
The bounded quasi-periodic relative trajectories are investigated in this paper for on-orbit surveillance, inspection or repair, which requires rapid changes in formation configuration for full three-dimensional imaging and unpredictable evolutions of relative trajectories for non-allied spacecraft. A linearized differential equation for modeling J 2 perturbed relative dynamics is derived without any simplified treatment of full short-period effects. The equation serves as a nominal reference model for stationkeeping controller to generate the quasi-periodic trajectories near the equilibrium, i.e., the location of the chief. The developed model exhibits good numerical accuracy and is applicable to an elliptic orbit with small eccentricity inheriting from the osculating conversion of orbital elements. A Hamiltonian structure-preserving controller is derived for the three-dimensional time-periodic system that models the J 2-perturbed relative dynamics on a mean circular orbit. The equilibrium of the system has time-varying topological types and no fixed-dimensional unstable/stable/center manifolds, which are quite different from the two-dimensional time-independent system with a permanent pair of hyperbolic eigenvalues and fixed-dimensions of unstable/stable/ center manifolds. The unstable and stable manifolds are employed to change the hyperbolic equilibrium to elliptic one with the poles assigned on the imaginary axis. The detailed investigations are conducted on the critical controller gain for Floquet stability and the optimal gain for the fuel cost, respectively. Any initial relative position and velocity leads to a bounded trajectory around the controlled elliptic equilibrium. The numerical simulation indicates that the controller effectively stabilizes motions relative to the perturbed elliptic orbit with small eccentricity and unperturbed elliptic orbit with arbitrary eccentricity. The developed controller stabilizes the quasi-periodic relative trajectories involved in six foundational motions with different frequencies generated by the eigenvectors of the Floquet multipliers, rather than to track a reference relative configuration. Only the relative positions are employed for the feedback without the information from the direct measurement or the filter estimation of relative velocity. So the current controller has potential applications in formation flying for its less computation overload for on-board computer, less constraint on the measurements, and easily-achievable quasi-periodic relative trajectories. 相似文献
15.
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. 相似文献
16.
A new second-order solution to the two-point boundary value problem for relative motion about orbital rendezvous in one orbit
period is proposed. First, nonlinear differential equations to describe the relative motion between a chaser and a target
are presented considering the second-order terms in the gravity. Then, by regarding the second-order terms as external accelerations,
we establish second-order state transition equations. Moreover, the J2 perturbations effects can also be considered in the
state transition equations. Last, the initial relative velocity to fulfill a rendezvous is determined by solving the state
transition equations. Numerical simulations show that the new second-order state transition equations are accurate. The second-order
solution to the two-point boundary value problem on eccentric orbits is valid even if the relative range is farther than 500 km. 相似文献
17.
J. C. van der Ha 《Celestial Mechanics and Dynamical Astronomy》1982,26(3):285-309
The three-dimensional relative motion of a subsatellite with respect to a reference station in an elliptical orbit is studied. A general theory based on the variation of the relative elements, i.e. the instantaneous differences between the orbital parameters of the subsatellite and those of the station, is formulated in order to incorporate arbitrary perturbing forces acting on both satellites. The loss of precision inherent in the subtraction of almost identical quantities is avoided by the consistent use of difference variables. In the absence of perturbations exact analytical representations can be obtained for the relative state parameters. The influences of air drag and Earth's oblateness on the relative motion trajectories are investigated and illustrated graphically for a number of cases. 相似文献
18.
Robert M. L. Baker Jr Neil H. Jacoby Jr 《Celestial Mechanics and Dynamical Astronomy》1977,15(2):137-160
It has long been recognized and demonstrated in the astrodynamic literature that three observations of angular position are not always sufficient to determine a preliminary orbit. One reason for this is due to the fact that as the plane of the observer's motion approaches the plane of the orbit of the observed object, the determination of the orbit of the object becomes indeterminant. Merely changing the coordinate system will not eliminate the inherent indeterminacy or singularity. When the observed object is moving in the same plane as the observer, their relative motion is described in two dimensions rather than three. The problem reduces to defining two components of position and two of velocity given only three angular measures and no solution is possible. Although this singularity is a rather old, albeit infrequently arising problem in celestial mechanics, it has received renewed interest due to the advent of satellite observatories that observe other spacecraft. In this new circumstance the plane of the observer's motion is rather frequently near the plane of the object (12% to 35% of the time) and the co-planar singularity becomes a subject that deserves additional attention.It is the purpose of this paper to develop a practical and simple method of orbit determination using four observations. This method also allows one to avoid the problem of multiple orbit-determination solution roots, and provides numerical indices that are useful in assessing the degree of indeterminacy in any given observer/object geometry. This paper does not dwell at length on the theory of orbital singularities, since they have been already treated in celestial mechanics literature. Instead, the emphasis is on the details of a new computational technique, which has been found to be computationally more efficient than previous four-observation methods, and which is unique in being formulated in the geocentric system and involves only one scalar quantity in the correction process.The equations for the new method are developed and a numerical example is presented that demonstrates the efficiency of the method. 相似文献
19.
This paper presents a Hamiltonian approach to modelling spacecraft motion relative to a circular reference orbit based on
a derivation of canonical coordinates for the relative state-space dynamics. The Hamiltonian formulation facilitates the modelling
of high-order terms and orbital perturbations within the context of the Clohessy–Wiltshire solution. First, the Hamiltonian
is partitioned into a linear term and a high-order term. The Hamilton–Jacobi equations are solved for the linear part by separation,
and new constants for the relative motions are obtained, called epicyclic elements. The influence of higher order terms and
perturbations, such as Earth’s oblateness, are incorporated into the analysis by a variation of parameters procedure. As an
example, closed-form solutions for J2-invariant orbits are obtained. 相似文献
20.
Claudio Bombardelli Juan Luis Gonzalo Javier Roa 《Celestial Mechanics and Dynamical Astronomy》2017,127(1):49-66
A compact, time-explicit, approximate solution of the highly non-linear relative motion in curvilinear coordinates is provided under the assumption of circular orbit for the chief spacecraft. The rather compact, three-dimensional solution is obtained by algebraic manipulation of the individual Keplerian motions in curvilinear, rather than Cartesian coordinates, and provides analytical expressions for the secular, constant and periodic terms of each coordinate as a function of the initial relative motion conditions or relative orbital elements. Numerical test cases are conducted to show that the approximate solution can be effectively employed to extend the classical linear Clohessy–Wiltshire solution to include non-linear relative motion without significant loss of accuracy up to a limit of 0.4–0.45 in eccentricity and 40–45\(^\circ \) in relative inclination for the follower. A very simple, quadratic extension of the classical Clohessy–Wiltshire solution in curvilinear coordinates is also presented. 相似文献