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.     

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.     

航天飞行器相对运动动力学及其控制方法

采用本地轨道从标系对两邻飞船间相对运动动力学展开研究，指出在相对运动中也存在平衡状态，用相平面方法分析了其稳定性，基于此分析，综合出相对运动控制方法，即距离速率控制方法，受控运动轨迹是一条稳定的稳态直线，进而建立了全方位距离速率控制方法。最后以系绳卫星系统和飞船安全为例完成了计算机模拟。     

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.     

相邻两航天器相对运动近似方程的解析解及比动能方程

在地心引力场中,当目标航天器沿近圆轨道作无动力运动时,与目标航天器相邻的受控航天器相对于目标航天器的运动可以近似地用Hill方程描述。文章给出了受控航天器对目标航天器运动的推力加速度随时间线性变化时Hill方程的解析解。并根据Hill方程导出了受控航天器相对目标航天器运动的比动能方程。还讨论了比动能方程在上述两航天器轨道相遇和轨道交会问题中的应用。     

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.     

空间最优交会

在相对体制下,本文讨论了混合指标下的最优交会问题。除给出了最优交会体制导律的结构形式外,还提供了求解最优交会制导律的参数方程组。该方程具有迭代求解形式,给数字求解带来极大方便。     

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.     

极坐标系连续常值推力机动分析

连续常值推力是空间飞行常用的轨道机动方式，在空间交会与星际航行使命中具有重要的应用价值。其中，小推力适合于地球轨道航天器交会机动，而切向或周向推力以及较大的正径向推力可用于脱离地球引力场的逃逸飞行，执行星际交会使命。应用常推力作用下的极坐标系质心运动方程，对机动推力的量值没有限制；在航天器交会应用中，对相对距离也无要求。这种方法可直接获得向径与速度等轨道参数随时间或极角（绕地心的转动角）的变化，便于分析轨道转移与逃逸运动，有助于飞行使命与运动轨迹的设计。特别是，若机动转移的初轨为圆轨道，在推力较小、飞行时间不长的情况下，应用无量纲形式运动方程，可获得具有工程应用价值的近似解。文章给出一些有关的结果与应用案例。     

Optimal control of the deployment process of solar wings on spacecraft

The optimal attitude control problem of spacecraft during the stretching process of solar wings is investigated in this paper. The dynamical equations of the nonholonomic system are derived from the conservation principle of the angular momentum of the multibody system. Attitude control of the spacecraft with internal motion is reduced to a nonholonomic motion planning problem. The spacecraft attitude control is transformed into the steering problem for a drift free control system. The optimal solution for steering a spacecraft with solar wings is presented. The controlled motion of spacecraft is simulated for two cases. The numerical results demonstrate the effectiveness of the optimal control approach.     

六自由度气浮台及其姿态平台自动平衡系统设计

针对卫星交会对接中相对姿态和位置的控制问题,设计一种6自由度（DOF）气浮台,并推导出描述两个气浮台模拟卫星交会对接运动的耦合6自由度(DOF)动力学模型。为解决气浮台模拟空间失重环境中的重力力矩干扰问题,设计一种基于应用新颖的快速非奇异终端滑模面的自适应有限时间控制器的自动平衡系统,估计出质心位置,通过控制三个平衡质量块补偿姿态平台质心与旋转中心之间的位置偏差。李雅普诺夫理论推导和仿真结果表明,该方法可以快速完成对姿态平台质心位置的准确估计,实现质心与旋转中心的高度重合。     

System test results from the GNC experiments on the PRISMA in-orbit test bed

The PRISMA in-orbit test bed will demonstrate guidance, navigation, and control strategies for spacecraft formation flying and rendezvous. The project is funded by the Swedish National Space Board and the prime contractor is the Swedish Space Corporation. The project is further supported by the German Aerospace Center, the Technical University of Denmark, and the French Space Agency. PRISMA was launched on June 15, 2010 and after three weeks of operations, all on-board systems and units have passed an initial commissioning phase. Separation of the two PRISMA satellites from each other is expected by mid-August 2010.PRISMA consists of two spacecraft: MAIN and TARGET. The MAIN spacecraft has full orbit control capability while TARGET is attitude controlled only.The Swedish Space Corporation is responsible for three groups of guidance, navigation, and control experiments. These experiments include GPS- and vision-based formation flying during which the spacecraft will fly in passive as well as forced motion. The three experiments are: autonomous formation flying, proximity operations with final approach/recede maneuvers, and autonomous rendezvous. This paper presents system test results from two of these experiments as obtained with the flight-ready system. The system tests consist of a series of simulations performed on the flight model spacecraft with a large amount of hardware in the loop.     

Experimental investigation of the modes of operation of uncontrolled attitude motion of the Progress spacecraft

Results of in-flight tests of three modes of uncontrolled attitude motion of the Progress spacecraft are described. These proposed modes of experiments related to microgravity are as follows: (1) triaxial gravitational orientation, (2) gravitational orientation of the rotating satellite, and (3) spin-up in the plane of the orbit around the axis of the maximum moment of inertia. The tests were carried out from May 24 to June 1, 2004 onboard the spacecraft Progress M1-11. The actual motion of this spacecraft with respect to its center of mass, in the above-mentioned modes, was determined by telemetric information about an electric current tapped off from solar batteries. The values of the current obtained during a time interval of several hours were processed jointly using the least squares method by integration of the equations of the spacecraft’s attitude motion. The processing resulted in estimation of the initial conditions of motion and of the parameters of mathematical models used. For the obtained motions the quasi-static component of microaccelerations was computed at a point onboard, where installation of experimental equipment is possible.     

Modes of uncontrolled rotational motion of the <Emphasis Type="Italic">Progress M-29M</Emphasis> spacecraft

We have reconstructed the uncontrolled rotational motion of the Progress M-29M transport cargo spacecraft in the single-axis solar orientation mode (the so-called sunward spin) and in the mode of the gravitational orientation of a rotating satellite. The modes were implemented on April 3–7, 2016 as a part of preparation for experiments with the DAKON convection sensor onboard the Progress spacecraft. The reconstruction was performed by integral statistical techniques using the measurements of the spacecraft’s angular velocity and electric current from its solar arrays. The measurement data obtained in a certain time interval have been jointly processed using the least-squares method by integrating the equations of the spacecraft’s motion relative to the center of mass. As a result of processing, the initial conditions of motion and parameters of the mathematical model have been estimated. The motion in the sunward spin mode is the rotation of the spacecraft with an angular velocity of 2.2 deg/s about the normal to the plane of solar arrays; the normal is oriented toward the Sun or forms a small angle with this direction. The duration of the mode is several orbit passes. The reconstruction has been performed over time intervals of up to 1 h. As a result, the actual rotational motion of the spacecraft relative to the Earth–Sun direction was obtained. In the gravitational orientation mode, the spacecraft was rotated about its longitudinal axis with an angular velocity of 0.1–0.2 deg/s; the longitudinal axis executed small oscillated relative to the local vertical. The reconstruction of motion relative to the orbital coordinate system was performed in time intervals of up to 7 h using only the angularvelocity measurements. The measurements of the electric current from solar arrays were used for verification.     

基于双目视觉的相对状态自主确定

基于双目视觉系统提出了一种适于交会对接最后阶段的近距离绕飞的相对状态自主确定方法。用修正罗德里格参数(MRP)描述姿态,给出了追踪器与目标器间的相对姿态运动学与动力学方程,以及相对运动方程,根据双目视觉系统的小孔成像原理建立了双目系统的测量数学模型,由扩展卡尔曼滤波(EKF)获得两航天器的相对位姿及其变化率。仿真结果表明该法可行。     

航天员舱外相对航天器运动轨迹摄动研究

用轨道根数描述载人航天器运动,在其轨道坐标系(LVLH)中,建立了含地球J2项引力和大气阻力摄动加速度的航天员质心相对航天器的运动模型。在参考轨道存在小偏心率时,对基于圆参考轨道假设推导的航天器编队飞行的线性时变系统状态矩阵进行了修正。仿真结果表明:对偏心率为0.002的航天器轨道,修正后模型所得航天员相对轨迹摄动量在离舱300 s后达到0.8%,是修正前模型的5倍,但计算量增加不超过5%,更适于航天员舱外飞行和近距离编队飞行建模。     

On the planar inertial motion of a system of three coupled bodies

Numerous papers are devoted to studying the motion of a system (coupling) of two bodies in the Earth’s satellite orbit ([1–4] and others). The problem on the planar inertial motion of three bodies, coupled by a non-extensible weightless string having the form of an unfastened chain, is considered in the paper. Such a configuration can be represented, for example, by a system of two coupled spacecraft rotating around their common center of mass (in order to simulate the gravity force) in long-term space missions, when the third body (the lift) is located on a connecting cable. The bodies are considered to be the material points (particles).     

一类椭圆轨道交会中的动力学方程研究

研究了一类追踪器和目标器轨道半长轴相差不大、轨道面外的距离相差不大的小偏心率椭圆交会的动力学问题.首先选择合适的圆轨道上的点建立参考系,推导出针对圆轨道参考系的无量纲化线性定常方程,并获得相应的相对状态;接着讨论该方程在小偏心椭圆轨道两冲量交会中的应用;最后进行数值仿真,验证动力学方程和制导策略,并与CW方程及制导策略的相关仿真进行比较.仿真结果表明本文给出的动力学方程的精度优于CW方程,能有效解决这类椭圆交会问题.     

Control of the deployment of an orbital tether system that consists of two small spacecraft

Control of an orbital tether system that consists of two small spacecraft has been considered. The proposed control laws are based on the modification of well-known programs for the deployment of tether system systems under the assumption that the masses of spacecraft and the tether are comparable in magnitude. To construct nominal deployment programs, we have developed a mathematical model of the motion of the given system in an orbital moving coordinate system taking into account the specific features of this problem. The performance of the proposed deployment programs is assessed by a mathematical model of the orbital tether system with distributed parameters written in the geocentric coordinate system. The test calculations involve a linear regulator that implements feedback on the tether length and velocity.     

Periodic motions of a nearly dynamically symmetric satellite in the neighborhood of hyperboloidal precession

The motion of a satellite close to a dynamically symmetric solid body in a Newtonian gravitational field over a circular orbit is studied. The system of differential equations describing the body’s motion is close to a system with cyclic coordinate. New classes of periodic motions are constructed in the neighborhood of a known partial solution to an unperturbed problem, hyperboloidal precession of a dynamically symmetric satellite. In the resonance case, when the ratio of one frequency of small oscillations of a reduced system with two degrees of freedom in the neighborhood of a stable equilibrium position to the frequency of cyclic coordinate variation is close to an integer number, there exist one or three families of periodic motions that are analytic in terms of fractional powers of a small parameter. A study of stability of these motions was performed with the help of KAM (Kolmogorov-Arnold-Moser) theoty. Faling the described resonance there exists a unique family of periodic motions that is analytic in terms of integer powers of a small parameter. The check-up of stability of these motrons was carried out. We distinguished the cases of parametric resonance, resonances of the third and fourth orders, and a non-resonant case. In the resonance cases our study relies on well-known results on stability of Hamiltonian systems during resonances [1]. In the non-resonant case we use the KAM theory [2].