Small spacecraft formation using potential functions

A group of small spacecraft able to change its orbital formation through using the potential function is discussed. Spacecraft shapes, sizes, and maneuvering capabilities in general are not identical. All objects are assumed to maneuver under discrete thruster effects. A hyperbolic form of attractive potential function is then used to reduce the control intervention by using the natural orbital motion for approaching goal configuration. A superquadric repulsive potential with 3D rigid object representation is then used to have more accurate mutual sensing between objects. As spacecraft start away from their goals, the original parabolic attractive potential becomes inefficient as the continuous control force increases with distance linearly. The hyperbolic attractive potential offers good representation of the control force independent of the distance to goal, ensuring global stability as well.     

Small spacecraft formation using potential functions

A group of small spacecraft able to change its orbital formation through using the potential function is discussed. Spacecraft shapes, sizes, and maneuvering capabilities in general are not identical. All objects are assumed to maneuver under discrete thruster effects. A hyperbolic form of attractive potential function is then used to reduce the control intervention by using the natural orbital motion for approaching goal configuration. A superquadric repulsive potential with 3D rigid object representation is then used to have more accurate mutual sensing between objects. As spacecraft start away from their goals, the original parabolic attractive potential becomes inefficient as the continuous control force increases with distance linearly. The hyperbolic attractive potential offers good representation of the control force independent of the distance to goal, ensuring global stability as well.     

The Use of Quaternions in the Optimal Control Problems of Motion of the Center of Mass of a Spacecraft in a Newtonian Gravitational Field: III

The results of numerical solution of the problem of a rendezvous in the central Newtonian gravitational field of a controlled spacecraft with an uncontrollable spacecraft moving along an elliptic Keplerian orbit are presented. Two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems and using quaternion variables to describe the orientations of these coordinate systems. The problem of a rendezvous of two spacecraft is formulated [1, 2] as a problem of optimal control by the motion of the center of mass of a controlled spacecraft with a movable right end of the trajectory, and it is solved on the basis of Pontryagin's maximum principle. The paper is a continuation of papers [1, 2], where the problem of a rendezvous of two spacecraft has been considered theoretically using the two above variants of the equations of motion for the center of mass of the controlled spacecraft.     

Optimization of the trajectory of the spacecraft insertion into the system of heliocentric orbits

The possibility of the spacecraft insertion into the system of operational heliocentric orbits has been analyzed. It has been proposed to use a system of several operational heliocentric orbits. On each orbit, the spacecraft makes one or more revolutions around the Sun. These orbits are characterized by a relatively small perihelion radius and relatively high inclination, which allows one to investigate the polar regions of the Sun. The transition of the spacecraft from one orbit to another has been performed using an unpowered gravity assist maneuver near Venus and does not require the cruise propulsion operation. Each maneuver transfers the spacecraft into the sequence of operational heliocentric orbits. We have analyzed several systems of operational heliocentric orbits into which the spacecraft can be inserted by means of the considered transportation system with electric propulsion (EP). The mass of the spacecraft delivered to these systems of operational orbits has been estimated.     

One optimization problem for trajectories of spacecraft rendezvous mission to a group of asteroids

The optimization problem is considered for the trajectory of a spacecraft mission to a group of asteroids. The ratio of the final spacecraft mass to the flight time is maximized. The spacecraft is controlled by changing the value and direction of the jet engine thrust (small thrust). The motion of the Earth, asteroids, and the spacecraft proceeds in the central Newtonian gravitational field of the Sun. The Earth and asteroids are considered as point objects moving in preset elliptical orbits. The spacecraft departure from the Earth is considered in the context of the method of a point-like sphere of action, and the excess of hyperbolic velocity is limited. It is required sequentially to have a rendezvous with asteroids from four various groups, one from each group; it is necessary to be on the first three asteroids for no less than 90 days. The trajectory is finished by arrival at the last asteroid. Constraints on the time of departure from the Earth, flight duration, and final mass are taken into account in this problem.     

Analysis of the orbital motion of the asteroid Apophis’ satellite

We have analyzed the orbital disturbed spacecraft motion near an asteroid. The equations of the asteroidocentric spacecraft motion have been used with regard to three perturbations from celestial bodies, the asteroid’s nonsphericity, and solar radiation pressure. It has been shown that the orbital parameters of the main spacecraft and a small satellite with a radio beacon can be selected such that the orbits are rather stable for a fairly long period of time, i.e., a few weeks for the main spacecraft with an orbit initial radius of ~0.5 km and a few years before approaching Apophis with the Earth in 2029, for a small satellite at an orbit initial radius of ~1.5 km. The initial orientation of the spacecraft orbital plane perpendicular to the sunward direction is optimal from the point of view of the stability of the spacecraft flight near an asteroid.     

On plane oscillations of a pendulum with variable length suspended on the surface of a planet’s satellite

The problem of planar oscillations of a pendulum with variable length suspended on the Moon’s surface is considered. It is assumed that the Earth and Moon (or, in the general case, a planet and its satellite, or an asteroid and a spacecraft) revolve around the common center of mass in unperturbed elliptical Keplerian orbits. We discuss how the change in length of a pendulum can be used to compensate its oscillations. We wrote equations of motion, indicated a rule for the change in length of a pendulum, at which it has equilibrium positions relative to the coordinate system rotating together with the Moon and Earth. We study the necessary conditions for the stability of these motions. Chaotic dynamics of the pendulum is studied numerically and analytically.     

Calculation and Study of Limited Orbits around the <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> Libration Point of the Sun–Earth System

A procedure has been proposed for calculating limited orbits around the L₂ libration points of the Sun–Earth system. The motion of a spacecraft in the vicinity of the libration point has been considered a superposition of three components, i.e., decreasing (stable), increasing (unstable), and limited. The proposed procedure makes it possible to correct the state vector of the spacecraft so as to neutralize the unstable component of the motion. Using this procedure, the calculation of orbits around various types of libration points has been carried out and the dependence on the orbit type on the initial conditions has been studied.     

On stability of a rotating spacecraft partially filled with liquid: The case of a single cavity

The problem of stability of a rotating spacecraft with a cavity partially filled with liquid to a small depth is considered with regard to the distinction in angular velocities of spacecraft and liquid rotation and their variability (the modes of the spacecraft’s stationary rotation, spin-up, and rotation deceleration). The regions of stability (in space of the characteristic parameters of an object) are found, and mathematical simulating of the disturbed motion is carried out.     

Spacecraft Injection into a Heliocentric Terrestrial Orbit

We consider the problem of injection of a spacecraft into the heliocentric Earth's orbit ahead and/or behind the Earth by 60° and 120° in heliographic longitude. The range of solar and astrophysical problems for which these orbits are necessary is reviewed. The variants of injection into heliocentric orbits work from a low around-Earth orbit with one turn-on of the engine in this orbit and one turn-on at the end of the injection trajectory. In this case, it turns out to be more profitable to put spacecraft into orbit for three or even four revolutions of the Earth about the Sun. The velocities necessary for the start from a low around-Earth orbit, the velocities at the final point of injection, and the fuel mass (relative to the spacecraft mass) necessary for injection are estimated. The problems for which injection to similar orbits is executed, using the low-thrust engine and with a combined regime of injection, are also considered.     

Calculation of quasisynchronous orbits of a spacecraft around Phobos for solving the problem of landing on its surface

The problem of selecting quasi-synchronous orbits of a spacecraft around Phobos is considered. These quasi-synchronous orbits are far (with respect to the Hill’s sphere) quasi-satellite orbits with retrograde rotation in the restricted three body problem. The orbit should pass through a given point at a specified time instant. It should also possess a property of minimum distance from the Phobos surface at every passage above the region of planned landing. The equations of dynamics are represented in the form describing the orbit as a combination of motions in two drifting ellipses, inner and outer ellipses. The center of the outer ellipse is located on the inner ellipse. A formula is derived that relates averaged values of half-axes of the inner and outer ellipses. It is used for construction of the first approximation of numerically designed orbit, which makes it possible to simplify and speed up the computing process. The tables of initial conditions obtained as a result of calculations are presented.     

Optimization of the Flight of a Spacecraft with a Solar Sail from the Earth to Mars with a Perturbation Maneuver near Venus

The trajectories of the fastest flight of a spacecraft (SC) with a solar sail from the Earth's sphere of activity to the Martian sphere of activity including the section of a perturbation maneuver near Venus are investigated. The planetary spheres of activity are assumed to be point-like; i.e., the maneuver section and the initial and final positions of the SC coincide with the corresponding positions of the planets. The initial velocity of the SC is assumed to be equal to the Earth's velocity, so that no leveling of the velocities of the SC and Mars in the final point of the flight is required. The perturbation maneuver is considered as a jump of the heliocentric velocity of the SC at the point of its contact with Venus, which does not change the magnitude of its Venus-centric velocity. The orbits of planets are assumed to be circular and coplanar; the SC trajectory lies at the plane of these orbits. The sail is planar with a specularly reflecting surface. The trajectories of optimum flights are determined as a result of solving the boundary value problem of the Pontryagin maximum principle. The families of solutions to this problem depending on the initial angular positions of Venus and Mars are constructed by the method of continuation over a parameter.     

Estimation of the Probability of Capture into a Resonance Mode of Motion for a Spacecraft during Its Descent in the Atmosphere

The estimation of the probability of capture into a resonance mode of motion is considered for a spacecraft with a small asymmetry during its entry into the atmosphere. It is assumed that the initial conditions of spacecraft motion are distributed uniformly in some sufficiently small domain. The problem is solved for the equations of spacecraft motion linear with respect to the angle of attack. An analytical estimate of the probability of the spacecraft capture into the resonance corresponding to an ascending branch of the velocity head is obtained. The emphasis in the analysis of the estimate is made on the effect of the spacecraft asymmetry type on the probability of capture. A comparison of the estimate with the results of numerical computation is carried out. A model problem concerning the construction of the domain of the spacecraft center of mass locations, most dangerous from the point of view of the realization of the stable resonant modes of motion, is solved.     

On the features of long-term evolution of a high-apogee orbit of the SPECTR-R spacecraft

The practical tasks related to qualitative investigation of long-term evolution of high-apogee orbits of artificial Earth satellites (AES), for which the main perturbing factors are gravitational perturbations from the Moon and the Sun, are considered. Attention is given to the problem of the ballistic lifetime of similar orbits, and the issues associated with possibilities of the correction of orbits for ensuring the required duration of their ballistic lifetime are considered. The orbit of the SPECTR-R spacecraft launched in July of 2011 is considered as an example.     

Time-optimal maneuvering control of a rigid spacecraft

The time-optimal rest-to-rest maneuvering control problem of a rigid spacecraft is studied in this paper. By utilizing an iterative procedure, this problem is formulated and solved as a constrained nonlinear programming (NLP) one. In this novel method, the count of control steps is fixed initially and the sampling period is treated as a variable in the optimization process. The optimization object is to minimize the sampling period below a specific minimum value, which is set in advance considering the accuracy of discretization. To generate initial feasible solutions of the NLP problem, a genetic-algorithm-based is also proposed such that the optimization process can be started from many different points to find the globally optimal solution. With the proposed method, one can find a time-optimal rest-to-rest maneuver of the rigid spacecraft between two attitudes. To show the feasibility of the proposed method, simulation results are included for illustration.     

单航天器无需机动对星座多星交会轨道全解

文献[1,2]已研究了单航天器无需变轨对Walker星座多星交会的轨道设计,在此基础上,依据交会点必是轨道交点的轨道特性,提出了轨道全解的解析法,可给出该轨道所能够交会卫星的最大数目,研究结果可为单航天器无需大机动变轨对星座多星接近的轨道应用提供理论参考。     

航天器新型非奇异饱和终端滑模姿态控制

针对刚体航天器姿态机动过程中存在的控制饱和与外部干扰问题,提出一类基于新型非奇异饱和终端滑模面的有限时间控制器设计方法。该控制方案不仅保证姿态机动过程的快速性,而且避免了传统的终端滑模面所带来的奇异性问题。此外,本文建议的控制器不仅显式考虑执行器输出力矩的饱和幅值要求,使航天器在饱和幅值的约束下完成姿态控制任务,而且控制器的设计对外部干扰的上界没有任何要求,也无需作任何小角度的假设。进一步的稳定性分析表明,通过引入新型非奇异饱和终端滑模,该控制器使得闭环系统能够快速收敛到滑模面的微小邻域内,继而收敛到平衡点的微小邻域内,并且系统对外部干扰具有较强的鲁棒性。数值仿真校验了该控制器在姿态机动过程中的性能。     

Optimization of Flights between the Orbits of Artificial Satellites of Two Planets Using a Combination of High and Low Thrusts

The problem of optimization of the trajectory of an interplanetary flight of a multistaged spacecraft using jet engines with high and low thrust is considered. The issues concerning the problem of choosing the main design parameters of a multistage spacecraft are touched upon. A mathematical model of start-to-finish optimization of all segments of the interplanetary flight trajectory is proposed. Using this model the specific features of flights to the orbits of satellites of Jupiter and Mars are studied.     

一类受周期扰动航天器的混沌姿态运动

研究了航天器从绕最小惯量主轴到最大惯量主轴旋转的姿态机动过程中的混沌现象。考虑到航天器内部或外部的振动部件的影响,假设两个主轴的转动惯量为时间的周期函数,同时还考虑了航天器内结构阻尼以及稀薄气体阻力的影响。应用高维的Melnikov方法,求解姿态机动过程中产生混沌的条件的解析表达式,且得到的阀值条件是扰动系统参数的函数。最后对该阀值条件进行了数值验证。