Optimal impulse point-to-elliptical orbit transfers of a spacecraft

The paper deals with energetically optimal multi-impulse transfer of a spacecraft in the central Newtonian gravity field near a planet. At the initial state of the transfer the distance from the spacecraft to the center of attraction, its radial and transversal velocity projections are known. At the end of the transfer the spacecraft must be located in the elliptical orbit with the given area and energy constants. The distance from the spacecraft to the center of attraction is bounded above and below, the transfer time being unspecified. The initial orbit intersects the inner boundary of the given ring.All the optimal solutions have been obtained by analytical way. A number of new solutions has been found for the given problem in comparison with the case of the transfer from the orbit at the free initial point.Up to five impulses can be applied on the optimal trajectories. The numerical simulation of the problem is carried out. It shows that all obtained solutions give not only local but global optimal energetic input on the corresponding conditions.     

Single-impulse transfers from orbits of artificial satellites to orbits around <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript> or <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> libration points

In the context of the restricted circular three-body problem a method for constructing families of periodic orbits is described. Each orbit contains a segment of transfer from artificial satellite orbit of a smaller body to an orbit around L ₁ or L ₂ points of the Sun-Earth and Earth-Moon systems, a segment of multiple flyby of this libration point, and a segment of return to the artificial satellite orbit. Dependences of velocities at the pericenter on the pericenter radius are given.     

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

The problem of optimal control is considered for the motion of the center of mass of a spacecraft in a central Newtonian gravitational field. For solving the problem, two variants of the equations of motion for the spacecraft center of mass are used, written in rotating coordinate systems. Both the variants have a quaternion variable among the phase variables. In the first variant this variable characterizes the orientation of an instantaneous orbit of the spacecraft and (simultaneously) the spacecraft location in this orbit, while in the second variant only the instantaneous orbit orientation is specified by it. The suggested equations are convenient in the respect that they allow the general three-dimensional problem of optimal control by the motion of the spacecraft center of mass to be considered as a composition of two interrelated problems. In the first variant these problems are (1) the problem of control of the shape and size of the spacecraft orbit and (2) the problem of control of the orientation of a spacecraft orbit and the spacecraft location in this orbit. The second variant treats (1) the problem of control of the shape and size of the spacecraft orbit and the orbit location of the spacecraft and (2) the problem of control of the orientation of the spacecraft orbit. The use of quaternion variables makes this consideration most efficient. The problem of optimal control is solved on the basis of the maximum principle. Several first integrals of the systems of equations of the boundary value problems of the maximum principle are found. Transformations are suggested that reduce the dimensions of the systems of differential equations of boundary value problems (without complicating them). Geometrical interpretations are given to the transformations and first integrals. The relation of the vectorial first integral of one of the derived systems of equations (which is an analog of the well-known vectorial first integral of the studied problem of optimal control) with the found quaternion first integral is considered. In this paper, which is the first part of the work, we consider the models of motion of the spacecraft center of mass that employ quaternion variables. The problem of optimal control by the motion of the spacecraft center of mass is investigated on the basis of the first variant of equations of motion. An example of a numerical solution of the problem is given.     

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

The problem of a rendezvous in the central Newtonian gravitational field is considered for a controlled spacecraft and an uncontrollable spacecraft moving along an elliptic Keplerian orbit. For solving the problem, 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. In the first variant of the equations of motion a quaternion variable characterizes the orientation of an instantaneous orbit of the spacecraft and the spacecraft location in the orbit, while in the second variant it characterizes the orientation of the plane of the spacecraft instantaneous orbit and the location of a generalized pericenter in the orbit. The quaternion variable used in the second variant of the equations of motion is a quaternion osculating element of the spacecraft orbit. The problem of a rendezvous of two spacecraft is formulated 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.     

About one problem of trajectory optimization

The optimization problem for trajectories of spacecraft flight from the Earth to an asteroid is considered in this paper. The flight is realized in the central Newtonian gravitational field of the Sun with a possibility of gravitational maneuvers near planets. Perturbation maneuvers are taken into account using the method of point area of action with a limitation on the flyby altitude. The spacecraft is controlled by changing the value and direction of the engine thrust. The problem is solved taking into account constraints on the launch time, flight duration, and minimum distance to the Sun.     

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.     

Solving Optimization Problems for the Flight Trajectories of a Spacecraft with a High-Thrust Jet Engine in Pulse Formulation for an Arbitrary Gravitational Field in a Vacuum

A mathematically well-posed technique is suggested to obtain first-order necessary conditions of local optimality for the problems of optimization to be solved in a pulse formulation for flight trajectories of a spacecraft with a high-thrust jet engine (HTJE) in an arbitrary gravitational field in vacuum. The technique is based on the Lagrange principle of derestriction for conditional extremum problems in a function space. It allows one to formalize an algorithm of change from the problems of optimization to a boundary-value problem for a system of ordinary differential equations in the case of any optimization problem for which the pulse formulation makes sense. In this work, such a change is made for the case of optimizing the flight trajectories of a spacecraft with a HTJE when terminal and intermediate conditions (like equalities, inequalities, and the terminal functional of minimization) are taken in a general form. As an example of the application of the suggested technique, we consider in this work, within the framework of a bounded circular three-point problem in pulse formulation, the problem of constructing the flight trajectories of a spacecraft with a HTJE through one or several libration points (including the case of going through all libration points) of the Earth–Moon system. The spacecraft is launched from a circular orbit of an Earth's artificial satellite and, upon passing through a point (or points) of libration, returns to the initial orbit. The expenditure of mass (characteristic velocity) is minimized at a restricted time of transfer.     

Guidance of a spacecraft with low lift-to-drag ratio to a landing point

The problem of terminal control over a deorbiting spacecraft at the stage of its flight after leaving plasma (altitude of ∼40 km) is considered, the aim being to guide it to a preset landing point. The algorithm is based on a modification of the well-known method of proportional navigation, when a fixed point is the target. It is suggested to use satellite navigation systems (of the GLONASS or GPS types) and/or radio beacons, which should allow one to determine the spacecraft trajectory parameters with high precision. Single-channel control is performed by changing the roll angle according to current parameters of the trajectory, which ensures adaptability of the method. Examples of three-dimensional trajectories of flight are presented for a manned spacecraft with low lift-to-drag ratio (∼0.5), currently under design in Russia. The results of statistical modeling taking into account initial deviations of the trajectory parameters and wind disturbances are presented. A method of statistical choice of a reference trajectory for the guidance stage is suggested. A theoretical possibility of using the algorithm of spacecraft guidance (in case of in-light accident with a carrier launcher) to preset regions in the vicinity of launching route is demonstrated. A qualitative analysis of proportional navigation with a fixed target is presented.     

The use of the Earth’s gravitational potential for interplanetary flights

An efficient scheme of the use of the Earth’s gravity in interplanetary flights is suggested, which opens up new opportunities for exploration of the solar system. The scheme of the gravitational maneuver allows one to considerably reduce the spacecraft mass consumption for a flight and the time of flight. An algorithm of the gravitational maneuver is suggested that takes into account the restriction on the altitude of a planet flyby. Estimates are made of transport capabilities for delivery of a spacecraft to the orbits of Jupiter, Saturn, and Uranus. The spacecraft is based on a middle-class carrier launcher of the Soyuz type and includes chemical and electric plasma jet engines of the SPD-140 type, which use solar energy.     

一种提高空间实验室定轨精度的方法

以近地航天器轨道动力学为基础,建立变阻力系数大气摄动模型,设计了求解变阻力系数的算法。然后利用天宫一号飞行器的测轨数据进行计算,分析了空间实验室飞行高度的轨道特性,其中包括：大气模式密度误差、变阻力系数与空间环境关系、定轨残差和星历误差。在空间环境平静和磁暴的条件下,制定了多种求解变阻力系数的策略,解决了空间实验室长弧段定轨精度受限的问题,并在空间环境平静条件下实现了优于10米的定轨精度,在磁暴条件下实现了优于20米的定轨精度。     

共面快速受控绕飞轨迹设计与控制

绕飞运动在航天器在轨服务与在轨支援、辅助航天员舱外活动、航天器编队飞行、空间交会对接等空间活动中具有重要应用。分析了快速受控绕飞的可行性和主要过程，建立了适用于目标航天器运行在圆轨道上的共面快速绕飞和进入绕飞与退出绕飞的轨迹设计模型，采用多速度脉冲控制方法和等角度，等时间控制方式对绕飞轨迹进行控制。仿真计算结果表明所提出的快速受控绕飞轨迹设计模型和控制方法可以实现对圆轨道目标航天器的共面快速受控绕飞。     

Optimal reconfiguration maneuvers for spacecraft imaging arrays in multi-body regimes

Upcoming National Aeronautics and Space Administration (NASA) mission concepts include satellite arrays to facilitate imaging and identification of distant planets. These mission scenarios are diverse, including designs such as the terrestrial planet finder-occulter (TPF-O), where a monolithic telescope is aided by a single occulter spacecraft, and the micro-arcsecond X-ray imaging mission (MAXIM), where as many as 16 spacecraft move together to form a space interferometer. Each design, however, requires precise reconfiguration and star tracking in potentially complex dynamic regimes. This paper explores control methods for satellite imaging array reconfiguration in multi-body systems. Specifically, optimal nonlinear control and geometric control methods are derived and compared to the more traditional linear quadratic regulators, as well as input state feedback linearization. These control strategies are implemented and evaluated for the TPF-O mission concept.     

Optimal reconfiguration maneuvers for spacecraft imaging arrays in multi-body regimes

Upcoming National Aeronautics and Space Administration (NASA) mission concepts include satellite arrays to facilitate imaging and identification of distant planets. These mission scenarios are diverse, including designs such as the terrestrial planet finder-occulter (TPF-O), where a monolithic telescope is aided by a single occulter spacecraft, and the micro-arcsecond X-ray imaging mission (MAXIM), where as many as 16 spacecraft move together to form a space interferometer. Each design, however, requires precise reconfiguration and star tracking in potentially complex dynamic regimes. This paper explores control methods for satellite imaging array reconfiguration in multi-body systems. Specifically, optimal nonlinear control and geometric control methods are derived and compared to the more traditional linear quadratic regulators, as well as input state feedback linearization. These control strategies are implemented and evaluated for the TPF-O mission concept.     

Robust algorithm of control for a maneuver in the Martian atmosphere to insert a spacecraft into the planet’s satellite orbit

When inserting a satellite into an orbit around Mars with the use of aerodynamic drag, it is required to apply a robust algorithm which is capable of being adapted to the actual conditions of the planet’s atmosphere. We suggest a method of adaptation taking into account the specific features of the maneuver including descending and ascending legs of the trajectory. It is demonstrated that the algorithm is efficient when disturbances of the density of the Martian atmosphere increase by a factor of 2–3.     

Trajectory Control for a Spacecraft with an Inflatable Braking Device in the Martian Atmosphere

The possibility of using an inflatable braking device for controlled descent in the Martian atmosphere of large-capacity cargoes is analyzed. The most complicated version of the trajectory control problem is considered, namely, the injection of a spacecraft at hyperbolic velocity into a parking orbit after braking in the atmosphere.     

Characterization of the optimal mass problem for aerobraking tethers

In the aerobraking tether concept, a probe, connected to an orbiter by a long, thin tether, passes through the atmosphere of a target planet to provide a desired velocity change, while keeping the orbiter above the sensible atmosphere. In earlier work, simple analytic models have been developed which accurately describe the characteristics of the mass-optimal tether. In this paper these models are generalized so that design of the spacecraft and the aerobraking maneuver can be completely characterized by four independent parameters. By comparing the tether mass (e.g. for aerocapture) with the propellant mass required to capture the orbiter, we show that aerobraking tethers have a clear advantage for a wide range of maneuvers.     

Halo orbits in the vicinity of the L 2 libration point in the Sun-Earth system

Methods are proposed for constructing the orbits of spacecraft remaining for long periods of time in the vicinity of the L ₂ libration point in the Sun-Earth system (so-called halo orbits), and the trajectories of uncontrolled flights from low near-Earth orbits to halo orbits. Halo orbits and flight trajectories are constructed in two stages: A suitable solution to a circular restricted three-body problem is first constructed and then transformed into the solution for a restricted four-body problem in view of the real motions of the Sun, Earth, and Moon. For a halo orbit, its prototype in the first stage is a combination of a periodic Lyapunov solution in the vicinity of the L ₂ point and lying in the plane of large-body motion, with the solution for the linear second-order system describing small deviations of the spacecraft from this plane along the periodic solution. The desired orbit is found as the solution to the three-body problem best approximating the prototype in the mean square. The constructed orbit serves as a similar prototype in the second stage. In both stages, the approximating solution is constructed by continuation along a parameter that is the length of the approximation interval. Flight trajectories are constructed in a similar manner. The prototype orbit in the first stage is a combination of a solution lying in the plane of large-body motion and a solution for a linear second-order system describing small deviations of the spacecraft from this plane. The planar solution begins near the Earth and over time tends toward the Lyapunov solution existing in the vicinity of the L ₂ point. The initial conditions of both prototypes and the approximating solutions correspond to the spacecraft’s departure from a low near-Earth orbit at a given distance, perigee, and inclination.     

Controlling the motion of a spacecraft when approaching a large object of space debris

The problem of calculating the parameters of maneuvering a spacecraft as it approaches a large object of space debris (LOSD) in close near-circular noncoplanar orbits has been considered. In [1–4], the results of analyzing the problem of the flyby of the separated LOSD groups have been presented. It has been assumed that a collector spacecraft approaches the LOSD and captures it or it is inserted into the nozzle of a small spacecraft that has a proper propulsion system (PS). However, in these papers, the flight from one object to another was only analyzed and the problem of approaching to LOSD with a given accuracy was not considered. This paper is a supplement to the cycle of papers [1–4]. It is assumed that, the final stage of approaching the LOSD is implemented by maneuvering in many orbits (up to several dozens) with low-thrust engines, but the PS operating time is fairly small compared with the orbit period in order to make it possible to use impulse approximation in the calculations.     

Translational–Rotational Motion of a Viscoelastic Sphere in Gravitational Field of an Attracting Center and a Satellite

We study the translational–rotational motion of a planet modeled by a viscoelastic sphere in the gravitational fields of an immovable attracting center and a satellite modeled as material points. The satellite and the planet move with respect to their common center of mass that, in turn, moves with respect to the attracting center. The exact system of equations of motion of the considered mechanical system is deduced from the D'Alembert–Lagrange variational principle. The method of separation of motions is applied to the obtained system of equations and an approximate system of ordinary differential equations is deduced which describes the translational–rotational motion of the planet and its satellite, taking into account the perturbations caused by elasticity and dissipation. An analysis of the deformed state of the viscoelastic planet under the action of gravitational forces and forces of inertia is carried out. It is demonstrated that in the steady-state motion, when energy dissipation vanishes, the planet's center of mass and the satellite move along circular orbits with respect to the attracting center, being located on a single line with it. The viscoelastic planet in its steady-state motion is immovable in the orbital frame of reference. It is demonstrated that this steady-state motion is unstable.     

绳系卫星安全捕获策略下的释放最优控制

安全捕获是绳系卫星系统空间应用的一个重要拓展。考虑系绳质量、系统质心变化及状态、控制约束,采用基于Lagrange方程的圆轨道条件下空间绳系网捕系统三维动力学模型。建立了安全捕获模型,推导得到零相对速度条件下的安全捕获末端条件。为保证方法的适用性,将基于Legendre伪谱法的连续时间最优控制问题离散为标准的非线性动态规划问题。最后在考虑释放控制前初始面外角偏差为5°的情况下,通过数值仿真验证了方法的有效性。仿真结果表明：对于远距离释放条件下的安全捕获,系统质心变化不容忽视;最小能量与绳长加速度约束下的控制响应相比最小时间与面内角约束要更平滑。