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: 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.     

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.     

Evolution of the Motion of a Viscoelastic Sphere in a Central Newtonian Field

The evolution of translational-rotational motion of a viscoelastic sphere in a central Newtonian field is studied. By the method of separating motions and averaging in the generalized Andoyer–Delaunay variables, equations are derived that describe the evolution of motion. The analysis of the approximate equations obtained is performed. The condition of existence of a steady-state motion is found, and its stability is investigated using the equations in variations.     

The Motion of a Material Point in a Newtonian Field under the Action of Equiangular Thrust

The problems of investigation and optimization of the motion of spacecraft are extensively discussed in the literature. Nevertheless, in many cases a large variety of qualitative characteristics of their motion and of the form of their trajectories are still unclear. In this paper we consider a plane equiangular acceleration of a spacecraft both in a Newtonian field and in its absence (at a large distance from the center of attraction). The general equation of a trajectory of plane acceleration is presented with the introduction of a new variable, an index of an exponent, which allows one to obtain convenient solutions at different values of the time-independent angle of inclination of the vector of thrust to the spacecraft's radius vector (i.e., when equiangular acceleration takes place). Asymptotic solutions are constructed and an interesting fact is revealed. Namely, it is shown that when the center of attraction exists or is absent, for all initial conditions the trajectories appearing at the above equiangular acceleration of a material point tend to the standard logarithmic spirals at a large distance from the center. Specifically, when the value of transverse (perpendicular to the radius vector) thrust is constant, there appears a logarithmic spiral with an angle of inclination to the radius vector equal to 35.264°. Different forms of the trajectory of equiangular acceleration of spacecraft at a low thrust are also studied. The results obtained can be useful for the investigation and choice of optimum space trajectories.     

Conditions of the Maximum Principle in the Problem of Optimal Control over an Aggregate of Dynamic Systems and Their Application to Solution of the Problems of Optimal Control of Spacecraft Motion

The necessary first-order conditions of strong local optimality (conditions of maximum principle) are considered for the problems of optimal control over a set of dynamic systems. To derive them a method is suggested based on the Lagrange principle of removing constraints in the problems on a conditional extremum in a functional space. An algorithm of conversion from the problem of optimal control of an aggregate of dynamic systems to a multipoint boundary value problem is suggested for a set of systems of ordinary differential equations with the complete set of conditions necessary for its solution. An example of application of the methods and algorithm proposed is considered: the solution of the problem of constructing the trajectories of a spacecraft flight at a constant altitude above a preset area (or above a preset point) of a planet's surface in a vacuum (for a planet with atmosphere beyond the atmosphere). The spacecraft is launched from a certain circular orbit of a planet's satellite. This orbit is to be determined (optimized). Then the satellite is injected to the desired trajectory segment (or desired point) of a flyby above the planet's surface at a specified altitude. After the flyby the satellite is returned to the initial circular orbit. A method is proposed of correct accounting for constraints imposed on overload (mixed restrictions of inequality type) and on the distance from the planet center: extended (nonpointlike) intermediate (phase) restrictions of the equality type.     

Locally Optimal Laws of Control of a Spacecraft Motion with an Electrodynamic Engine at Circumterrestrial Orbits

The motion of a spacecraft under the action of Ampére force is investigated. This force arises as a result of the interaction of the current in a rigid rod, at whose ends the devices which close the electric circuit through the circumterrestrial plasma are located, with the magnetic field of the Earth. The locally optimal laws of control by this rod orientation are found, when one of the elements of the orbit gains the maximum increment under the action of this engine. Numerical estimates of the gained increments are obtained, and it is demonstrated that this engine is a promising tool for the execution of light spacecraft maneuvers at low circumterrestrial orbits.     

Stationary Motions of a Homogeneous Meteor Ring in the Gravitational Field of a Center

The problem of studying a ring in the gravitational field of a center arose after the discovery of Saturn's rings by Galileo and subsequent discovery of the rings of other planets of the Solar System. Modern theoretical investigations of the existence and stability of planetary rings are mostly related to studies of plane differentially rotating discs [1]. As opposed to this line of research, this paper follows the approach established in classical works [2–4].     

Optimal Control of a Programmed Turn of a Spacecraft

A problem of optimal turn of a spacecraft is considered. The time of turn is minimized, as well as the functional having a meaning of the propellant consumption. An analytical solution to the problem stated is derived. It is demonstrated that the solution optimal in this sense belongs to a class of two-impulse controls, under which a spacecraft executes the turn along the trajectory of its free motion. The solution obtained in this paper differs from earlier available solutions considerably. The estimations of the propellant consumption for a realization of the programmed turn are made.     

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.     

Resonance Effects when Moving a Small Spacecraft around the Center of Mass in a Deployable Tether System

Cosmic Research - The resonance motions of a small spacecraft relative to the center of mass when deploying a tether system are analyzed. The tether system is deployed from a base spacecraft moving...     

椭圆参考轨道卫星编队构形的最优机动控制

研究了椭圆参考轨道编队飞行构形机动中的时间燃耗最优控制问题。以综合体现时间快速性和燃耗最优性的二次型性能指标作为优化准则,将构形机动的时间燃耗最优控制问题转化为一个线性规划问题,从而得到构形机动的最优机动时间以及相应的推力脉冲个数、幅值和作用时刻。最后在不计和计及地球引力摄动的两种不同情况下,分别利用Matlab和STK/Astrogator进行数值仿真,验证了控制算法的有效性。     

Optimal Control of a Spatial Turn of a Spacecraft with Minimal Loading of Construction

The problem of optimal control of a three-dimensional turn of a spacecraft is considered and solved. The turn is performed from an initial angular position into the required final angular position in a specified time and with a minimum value of the functional that represents the degree of loading of the construction. An analytical solution to the formulated problem is presented. It is demonstrated that the optimal (in this sense) control of the spacecraft reorientation can be determined in the class of a regular precession executed by the spacecraft. The instant when braking begins is determined based on the principles of terminal control using the actual kinematical parameters of the spacecraft motion, which substantially increases the accuracy of transferring the spacecraft to a specified position. Data of mathematical modeling are also presented that confirm the efficiency of the described method of controlling the spacecraft's three-dimensional turn.     

椭圆轨道欠驱动编队重构最优脉冲控制

针对椭圆轨道径向或迹向欠驱动编队重构问题,提出最优脉冲控制方法。首先,建立椭圆轨道欠驱动编队动力学模型,并基此分析径向或迹向欠驱动情况下的系统能控性与重构可行性。然后,解析推导两类欠驱动情况下实现编队重构所需的最少脉冲次数。基此,将椭圆轨道欠驱动编队重构最优脉冲控制问题表述为非线性规划问题,并采用遗传算法进行求解。最后,引入全驱动最优脉冲控制策略以及欠驱动最优连续控制策略进行对比分析,验证了欠驱动最优脉冲控制策略的有效性与正确性。仿真结果表明,欠驱动脉冲控制器可在缺失径向或迹向脉冲的条件下完成编队重构,并且保持与全驱动脉冲控制器类似的控制性能。     

接近和跟踪非合作机动目标的非线性最优控制

航天器在实施对空间非合作目标的近程操作任务中,需要接近目标并保持在目标附近的特定方位,对目标指定部位随动跟踪和观测。针对非合作机动目标的接近和视线跟踪的六自由度控制问题,根据视线坐标系下的相对轨道方程和体坐标系下的相对误差四元数姿态方程,建立了航天器间近距离相对运动的轨道和姿态联合控制模型。考虑模型的非线性、时变性和计算的快速性,采用θ-D控制方法进行接近和视线跟踪的轨道和姿态联合控制。为了减小跟踪同时存在轨道和姿态机动的非合作目标的控制误差,应用Lyapunov最小-最大定理设计了θ-D修正控制器,改善非合作目标同时进行姿态和轨道机动时的控制性能。仿真验证了模型的正确性和控制器良好的跟踪性能。     

A System of Stabilizing the Angular Motion of a Spacecraft Based on a Neuron-like System of Autonomous Adaptive Control

The aim of this paper is to apply the Autonomous Adaptive Control method to the problem of stabilization of a spacecraft's angular motion. The method is being developed at the Department of Simulation Systems of the Institute for System Programming, Russian Academy of Sciences. As applied to the problem under study, the main advantages of this method lie in the fact that there is no need to construct and incorporate into the control system a mathematical model of the controlled object, while it provides high-quality control, which cannot be ensured by a traditional control system. By optimizing the system with the use of genetic algorithms, one can save computational and hardware resources without losses in the quality of control and in the adaptive properties of the control system.     

Optimization of Flight Trajectories of a Spacecraft with Electric Propulsion Using the Gravitational Maneuver

Necessary conditions of optimality of the use of a gravitational maneuver during a flight are obtained, and a mathematical model for its study is proposed. With the help of the developed method of optimization of a trajectory of an interplanetary flight using a favorable gravitational maneuver, estimations of a spacecraft's transport capabilities are made for flights to Mercury and for the delivery of a solar probe into the near vicinity of the Sun.     

The Solution of Some Problems of Optimizing the Processes of a Flexible Correction of Motion of Satellite Systems: I

The first of a series of problems of the optimization of correction of satellite systems, moving over near-circular orbits, is considered. The correction is accomplished by means of low-thrust engines and is supposed to be flexible, where only the parameters of the relative motion of satellites must be corrected. The problem has a large dimension, but is invariant with respect to renumbering of satellites. This allows us to decompose the problem, i.e., to find new variables, linearly dependent on old ones, in which the problem breaks down into a series of independent subproblems of low dimension. The decomposition is accomplished by means of the technique [1] based on the theory of linear representations of groups.     

A Method of Semipassive Attitude Stabilization of a Spacecraft in the Geomagnetic Field

A method is proposed that enables one to accomplish semipassive attitude stabilization of a spacecraft moving in a circular Keplerian orbit in the geomagnetic field. The method is developed on the basis of the electrodynamic effect of the influence of the Lorentz forces acting on the charged spacecraft's surface. It possesses advantages such as control law simplicity, reliability, cost efficiency, small mass, and the possibility of using the basic control system components not only for attitude stabilization of a spacecraft but also for ensuring its electrostatic radiation screening. The possibility of implementing the method for slightly inclined orbits is proved analytically. Two versions of implementation of the method are proposed. The calculations confirmed the possibility of using also these versions for orbits whose inclinations are not small. The advantages of each version are revealed and practical recommendations for their utilization are given.