共查询到20条相似文献，搜索用时 421 毫秒
1.
The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the \(J_2\) effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the \(J_2\) perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a \(J_2\)-perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the \(J_2\) perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a \(J_2\)-perturbed reference orbit is also introduced. 相似文献
2.
SONG Ye-zhi HUANG Yong HU Xiao-gong LI Pei-jia CAO Jian-feng 《Chinese Astronomy and Astrophysics》2014
It is known that the dynamical orbit determination is the most common way to get the precise orbits of spacecraft. However, it is hard to build up the precise dynamical model of spacecraft sometimes. In order to solve this problem, the technique of the orbit determination with the B-spline approximation method based on the theory of function approximation is presented in this article. In order to verify the effectiveness of this method, simulative orbit determinations in the cases of LEO (Low Earth Orbit), MEO (Medium Earth Orbit), and HEO (Highly Eccentric Orbit) satellites are performed, and it is shown that this method has a reliable accuracy and stable solution. The approach can be performed in both the conventional celestial coordinate system and the conventional terrestrial coordinate system. The spacecraft's position and velocity can be calculated directly with the B-spline approximation method, it needs not to integrate the dynamical equations, nor to calculate the state transfer matrix, thus the burden of calculations in the orbit determination is reduced substantially relative to the dynamical orbit determination method. The technique not only has a certain theoretical significance, but also can serve as a conventional algorithm in the spacecraft orbit determination. 相似文献
3.
Jai Kumar 《Journal of Astrophysics and Astronomy》2018,39(6):66
In this study, the Chebyshev collocation method is used for solving the spacecraft relative motion of equations in Hill’s frame. Three different models of governing equations of relative motion (M1, M2, and M3) are considered and the maneuver cost required moving the spacecraft from one state to another is computed in the form of delta velocity at the first terminal point as a function of time of flight (TOF) and inter-satellite distance (ISD). A quantitative as well as qualitative difference is observed in the maneuver cost with the inclusion of radial and/or out of plane separation in along track separation of chaser. Also, a relative comparison of path profiles is made by considering M1, M2 and M3 models. Path profiles for M3 model are found close to M2 model for short intervals for a fixed ISD, whereas path profiles for M2 and M3 do not match even for small values of ISD for a fixed but long TOF. Path profiles for M1 models match to M2 model for very low values of target orbit eccentricities. 相似文献
4.
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
5.
Lambert problem solution in the hill model of motion 总被引：1，自引：0，他引：1
Alexander Sukhanov Antonio F. Bertachini A. Prado 《Celestial Mechanics and Dynamical Astronomy》2004,90(3-4):331-354
The goal of this paper is obtaining a solution of the Lambert problem in the restricted three-body problem described by the
Hill equations. This solution is based on the use of pre determinate reference orbits of different types giving the first
guess and defining the sought-for transfer type. A mathematical procedure giving the Lambert problem solution is described.
This procedure provides step-by-step transformation of the reference orbit to the sought-for transfer orbit. Numerical examples
of the procedure application to the transfers in the Sun–Earth system are considered. These examples include transfer between
two specified positions in a given time, a periodic orbit design, a halo orbit design, halo-to-halo transfers, LEO-to-halo
transfer, analysis of a family of the halo-to-halo transfer orbits. The proposed method of the Lambert problem solution can
be used for the two-point boundary value problem solution in any model of motion if a set of typical reference orbits can
be found. 相似文献
6.
Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy. 相似文献
7.
We analyze the out-of-plane librations of a tethered satellite system that is nominally rotating in the orbit plane. To isolate
the librational dynamics, the system is modeled as two point masses connected by a rigid rod with the system mass center constrained
to an unperturbed circular orbit. For small out-of-plane librations, the in-plane motion is unaffected by the out-of-plane
librations and a solution for the in-plane motion is determined in terms of Jacobi elliptic functions. This solution is used
in the linearized equation for the out-of-plane librations, resulting in a Hill’s equation. Floquet theory is used to analyze
the Hill’s equation, and we show that the out-of-plane librations are unstable for certain ranges of in-plane spin rate. For
relatively high in-plane spin rates, the out-of-plane librations are stable, and the Hill’s equation can be approximated by
a Mathieu’s equation. Approximate solutions to the Mathieu’s equation are determined, and we analyze the dominant characteristics
of the out-of-plane librations for high in-plane spin rates. The results obtained from the analysis of the linearized equations
of motion are compared to numerical simulations of the nonlinear equations of motion, as well as numerical simulations of
a more realistic system model that accounts for tether flexibility. The instabilities discovered from the linear analysis
are present in both the nonlinear system and the more realistic system model. The approximate solutions for the out-of-plane
librations compare well to the nonlinear system for relatively high in-plane rotation rates, and also capture the significant
qualitative behavior of the flexible system. 相似文献
8.
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
9.
Joseph W. Gangestad George E. Pollock James M. Longuski 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):125-145
A spacecraft that generates an electrostatic charge on its surface in a planetary magnetic field will be subject to a perturbative
Lorentz force. Active modulation of the surface charge can take advantage of this electromagnetic perturbation to modify or
to do work on the spacecraft’s orbit. Lagrange’s planetary equations are derived using the Lorentz force as the perturbation
on a Keplerian orbit, incorporating orbital inclination and true anomaly for the first time for an electrostatically charged
vehicle. The planetary equations reveal that orbital inclination is a second-order effect on the perturbation, explaining
results found in earlier studies through numerical integration. All of the orbital elements are coupled, but the coupling
notably does not depend on the magnitude of the electrostatic charge or on the strength of the magnetic field. Analytical
expressions that characterize this coupling are tested with a propellantless escape example at Jupiter. A closed-form solution
exists that constrains the set of equatorial orbits for which planetary escape is possible, and a sufficient condition is
identified for escape from inclined orbits. The analytical solutions agree with results from the numerically integrated equations
of motion to within a fraction of a percent. 相似文献
10.
Donald K. Yeomans 《Celestial Mechanics and Dynamical Astronomy》1996,66(1):1-12
To a significant degree, the success of spacecraft missions to comets and asteroids depends upon the accuracy of the target body ephemerides. In turn, accurate ephemerides depend upon the quality of the astrometric data set used in determining the object's orbit and the accuracy with which the target body's motion can be modelled. Using error analyses studies of the target bodies for the NEAR, Muses-C, Clementine 2, Stardust, and Rosetta missions, conclusions are drawn as to how to minimize target body position uncertainties at the times of encounter, In general, these uncertainties will be minimized when the object has a good number of optical observations spread over several orbital periods. If a target body lacks a lengthy data interval, its ephemeris uncertainties can be dramatically reduced with the use of radar Doppler and delay data taken when the body is relatively close to the Earth. The combination of radar and optical angle data taken at close Earth distances just before a spacecraft encounter can result in surprisingly small target body ephemeris uncertainties. 相似文献
11.
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
12.
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
13.
Marco Giancotti Stefano Campagnola Yuichi Tsuda Jun’ichiro Kawaguchi 《Celestial Mechanics and Dynamical Astronomy》2014,120(3):269-286
This work studies periodic solutions applicable, as an extended phase, to the JAXA asteroid rendezvous mission Hayabusa 2 when it is close to target asteroid 1999 JU3. The motion of a spacecraft close to a small asteroid can be approximated with the equations of Hill’s problem modified to account for the strong solar radiation pressure. The identification of families of periodic solutions in such systems is just starting and the field is largely unexplored. We find several periodic orbits using a grid search, then apply numerical continuation and bifurcation theory to a subset of these to explore the changes in the orbit families when the orbital energy is varied. This analysis gives information on their stability and bifurcations. We then compare the various families on the basis of the restrictions and requirements of the specific mission considered, such as the pointing of the solar panels and instruments. We also use information about their resilience against parameter errors and their ground tracks to identify one particularly promising type of solution. 相似文献
14.
《天文和天体物理学研究(英文版)》2017,(4)
A new non-simplified model of formation flying is derived in the presence of an oblate mainbody and third-body perturbation.In the proposed model,considering the perturbation of the thirdbody in an inclined orbit,the effect of obliquity(axial tilt) of the main-body is becoming important and has been propounded in the absolute motion of a reference satellite and the relative motion of a follower satellite.From a new point of view,J2 perturbed relative motion equations and considering a disturbing body in an elliptic inclined three dimensional orbit,are derived using Lagrangian mechanics based on accurate introduced perturbed reference satellite motion.To validate the accuracy of the model presented in this study,an auxiliary model was constructed as the Main-body Center based Relative Motion(MCRM) model.Finally,the importance of the main-body's obliquity is demonstrated by several examples related to the Earth-Moon system in relative motion and lunar satellite formation keeping.The main-body's obliquity has a remarkable effect on formation keeping in the examined in-track and projected circular orbit(PCO) formations. 相似文献
15.
V. A. Shefer 《Astronomy Letters》2006,32(12):845-858
The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion. 相似文献
16.
Alessandra F. S. Ferreira Antônio F. B. A. Prado Othon C. Winter Denilson P. S. Santos 《Astrophysics and Space Science》2018,363(7):145
Analytical equations describing the velocity and energy variation of a spacecraft in a Powered Swing-By maneuver in an elliptic system are presented. The spacecraft motion is limited to the orbital plane of the primaries. In addition to gravity, the spacecraft suffers the effect of an impulsive maneuver applied when it passes by the periapsis of its orbit around the secondary body of the system. This impulsive maneuver is defined by its magnitude \(\delta V\) and the angle that defines the direction of the impulse with respect to the velocity of the spacecraft (\(\alpha\)). The maneuver occurs in a system of main bodies that are in elliptical orbits, where the velocity of the secondary body varies according to its position in the orbit following the rules of an elliptical orbit. The equations are dependent on this velocity. The study is done using the “patched-conics approximation”, which is a method of simplifying the calculations of the trajectory of a spacecraft traveling around more than one celestial body. Solutions for the velocity and energy variations as a function of the parameters that define the maneuver are presented. An analysis of the efficiency of the powered Swing-By maneuver is also made, comparing it with the pure gravity Swing-by maneuver with the addition of an impulse applied outside the sphere of influence of the secondary body. After a general study, the techniques developed here are applied to the systems Sun-Mercury and Sun-Mars, which are real and important systems with large eccentricity. This problem is highly nonlinear and the dynamics very complex, but very reach in applications. 相似文献
17.
In this paper, we study the invariant manifold and its application in transfer trajectory problem from a low Earth parking orbit to the Sun-Earth \(L_{1}\) and \(L_{2}\)-halo orbits with the inclusion of radiation pressure and oblateness. Invariant manifold of the halo orbit provides a natural entrance to travel the spacecraft in the solar system along some specific paths due to its strong hyperbolic character. In this regard, the halo orbits near both collinear Lagrangian points are computed first. The manifold’s approximation near the nominal halo orbit is computed using the eigenvectors of the monodromy matrix. The obtained local approximation provides globalization of the manifold by applying backward time propagation to the governing equations of motion. The desired transfer trajectory well suited for the transfer is explored by looking at a possible intersection between the Earth’s parking orbit of the spacecraft and the manifold. 相似文献
18.
In this first part of the work we develop the equations of motion of a triaxial space station in orbit around the oblate Earth. A first order solution of the problem is presented and the method of complete integration of the system is outlined up to second order of approximation. The zero order part of the Hamiltonian includes both the kinetic and potential energy (Earth's Newtonian attraction) of the station, while the motion in the vicinity of a specific configuration is assumed.The solution leads to deviations on the attitude introduced by the oblateness of the Earth. Such attitude is an exact solution of the equations of the station when its center of mass moves in an elliptic Keplerian orbit.The explicit expressions of the complete solution, discussion of other possible effects on the motion and numerical comparisons will be presented in the second part of the work. 相似文献
19.
W. E. Williamson 《Celestial Mechanics and Dynamical Astronomy》1972,5(2):174-188
The perturbation method, a numerical method for solving two point boundary value problems (TPBVP), is modified to attempt to improve inherent instability and sensitivity problems associated with the method. The desired solution to the TPBVP is divided into two time intervals. The differential equations required to define a solution to the two point boundary value problem are integrated independently over these shorter segments rather than consecutively over the entire trajectory. The independent integration of the differential equations over approximately half of the trajectory instead of the entire trajectory substantially decreases sensitivity and stability properties associated with the numerical integration. The equations for both time segments can be integrated simultaneously. By this procedure, a system of twice the dimension of the original problem is integrated for a period of time equal to half of the time interval for the original problem. To show the effectiveness of the method, two impulse trajectories which minimize the total velocity increment required to transfer a spacecraft from an Earth orbit into a lunar orbit are calculated. 相似文献
20.
Elbert E. N. Macau 《Celestial Mechanics and Dynamical Astronomy》2003,87(3):291-305
The purpose of this work is to show that chaos control techniques (OGY, in special) can be used to efficiently keep a spacecraft around another body performing elaborate orbits. We consider a satellite and a spacecraft moving initially in coplanar and circular orbits, with slightly different radii, around a heavy central planet. The spacecraft, which is the inner body, has a slightly larger angular velocity than the satellite so that, after some time, they eventually go to a situation in which the distance between them becomes sufficiently small, so that they start to interact with one another. This situation is called as an encounter. In previous work we have shown that this scenario is a typical situation of a chaotic scattering for some well-defined range of parameters. Considering this scenario, we first show how it is possible to find the unstable periodic orbits that are located in the chaotic invariant set. From the set of unstable periodic orbits, we select the ones that can be combined to provide the desired elaborate orbit. Then, chaos control technique based on the OGY method is used to keep the spacecraft in the desired orbit. Finally, we analyze the results and make considerations regarding a realistic scenario of space exploration. 相似文献