A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits.This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rota...A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits.This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rotating frame with respect to the orbital elements.The resulted solution is expressed in terms of two different sets of orbital elements.The first one is the classical orbital elements and the second one is the nonsingular orbital elements.Among of them,however,the semi-latus rectum and true anomaly are used due to their generality,rather than the semi-major axis and mean anomaly that are used in most references.This specific selection for orbital elements yields a new solution that is universally applicable to elliptic,parabolic and hyperbolic orbits.It is shown that the new orbital element-based relative motion equations are equivalent to the Tschauner–Hempel equations.A linear map between the initial orbital element differences and the integration constants associated with the solution of the Tschauner–Hempel equations is constructed.Finally,the presented solution is validated through comparison with a high-fidelity numerical orbit propagator.The numerical results demonstrate that the new solution is computationally effective;and the result is able to match the accuracy that is required for linear propagation of spacecraft relative motion over a broad range of Keplerian orbits.展开更多
For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node...For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argu-ment of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guaran-tee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.展开更多
Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft ...Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.展开更多
The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dynamics in their design. In this work, the motion in invariant relati...The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dynamics in their design. In this work, the motion in invariant relative orbits is considered under the effects of second-order zonal harmonics in an equatorial orbit. The Hamiltonian framework is used to formulate the problem. All the possible conditions of the invariant relative motion are obtained with different inclinations of the follower satellite orbits. These second-order conditions warrantee the drift rates keeping two, or more, neighboring orbits from drifting apart. The conditions have been modeled. All the possibilities of choosing mean elements of the leader satellite orbit and differences in momenta between leader and follower satellites’ orbits are presented.展开更多
In recent years, there is a wide interest in Sarkovskii's theorem ami the related study. According to Sarkovskii's theoren if the continuous self-mapf of the closed interval has a 3-pcriodic orbit, then fmust ...In recent years, there is a wide interest in Sarkovskii's theorem ami the related study. According to Sarkovskii's theoren if the continuous self-mapf of the closed interval has a 3-pcriodic orbit, then fmust has an n-pcriodic orbit for any positive integer n. But f can not has all n-periodic orbits for some n.For example, letEvidently, f has only one kind of 3-periodic orbit in the two kinds of 3-periodic orbits. This explains that it isn't far enough to uncover the relation between periodic orbits by information which Sarkovskii's theorem has offered. In this paper, we raise the concept of type of periodic orbits, and give a feasible algorithm which decides the relation of implication between two periodic orbits.展开更多
基金This work was supported by the National Natural Science Foundation of China(Grant No.61403416)the“The Hundred Talents Program”of Chinese Academy of Science.
文摘A new formulation of the orbital element-based relative motion equations is developed for general Keplerian orbits.This new solution is derived by performing a Taylor expansion on the Cartesian coordinates in the rotating frame with respect to the orbital elements.The resulted solution is expressed in terms of two different sets of orbital elements.The first one is the classical orbital elements and the second one is the nonsingular orbital elements.Among of them,however,the semi-latus rectum and true anomaly are used due to their generality,rather than the semi-major axis and mean anomaly that are used in most references.This specific selection for orbital elements yields a new solution that is universally applicable to elliptic,parabolic and hyperbolic orbits.It is shown that the new orbital element-based relative motion equations are equivalent to the Tschauner–Hempel equations.A linear map between the initial orbital element differences and the integration constants associated with the solution of the Tschauner–Hempel equations is constructed.Finally,the presented solution is validated through comparison with a high-fidelity numerical orbit propagator.The numerical results demonstrate that the new solution is computationally effective;and the result is able to match the accuracy that is required for linear propagation of spacecraft relative motion over a broad range of Keplerian orbits.
基金the French government under the No de dossier: 688028B
文摘For a satellite in an orbit of more than 1600 km in altitude, the effects of Sun and Moon on the orbit can’t be negligible. Working with mean orbital elements, the secular drift of the longitude of the ascending node and the sum of the argu-ment of perigee and mean anomaly are set equal between two neighboring orbits to negate the separation over time due to the potential of the Earth and the third body effect. The expressions for the second order conditions that guaran-tee that the drift rates of two neighboring orbits are equal on the average are derived. To this end, the Hamiltonian was developed. The expressions for the non-vanishing time rate of change of canonical elements are obtained.
文摘Spacecraft formation flying is an attractive new concept in international aeronautic fields because of its powerful functions and low cost. In this paper, the formation design and PD closed-loop control of spacecraft formation flying in elliptical orbits are discussed. Based on two-body relative dynamics, the true anomaly is applied as independent variable instead of the variable of time. Since the apogee is considered as the starting point, the six integrating constants are calculated. Therefore, the algebraic solution is obtained for the relative motion in elliptical orbits. Moreover, the formation design is presented and both circular formation and line formation are provided in terms of an algebraic solution. This paper also discusses the PD-closed loop control for precise formation control in elliptical orbits. In this part, the error-type state equation is put forward and the linear quadratic regulator （LQR） method is used to calculate PD parameters. Though the gain matrix calculated from LQR is time-variable because the error-type state equation is time variable, the PD parameters are also considered as constants because of their small changes in simulation. Finally, taking circular formation as an example, the initial orbital elements are achieved for three secondary spacecraft. And the numerical simulation is analyzed under PD formation control with initial errors and J2 perturbation. The simulation results demonstrate the validity of PD closed-loop control scheme.
文摘The relative motion between multiple satellites is a developed technique with many applications. Formation-flying missions use the relative motion dynamics in their design. In this work, the motion in invariant relative orbits is considered under the effects of second-order zonal harmonics in an equatorial orbit. The Hamiltonian framework is used to formulate the problem. All the possible conditions of the invariant relative motion are obtained with different inclinations of the follower satellite orbits. These second-order conditions warrantee the drift rates keeping two, or more, neighboring orbits from drifting apart. The conditions have been modeled. All the possibilities of choosing mean elements of the leader satellite orbit and differences in momenta between leader and follower satellites’ orbits are presented.
基金Projects Supported by the National Natural Science Foundation of China
文摘In recent years, there is a wide interest in Sarkovskii's theorem ami the related study. According to Sarkovskii's theoren if the continuous self-mapf of the closed interval has a 3-pcriodic orbit, then fmust has an n-pcriodic orbit for any positive integer n. But f can not has all n-periodic orbits for some n.For example, letEvidently, f has only one kind of 3-periodic orbit in the two kinds of 3-periodic orbits. This explains that it isn't far enough to uncover the relation between periodic orbits by information which Sarkovskii's theorem has offered. In this paper, we raise the concept of type of periodic orbits, and give a feasible algorithm which decides the relation of implication between two periodic orbits.
