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.     

Optimal control of reorientation of a spacecraft using free trajectory method

The problem of optimal control over spatial reorientation of a spacecraft is considered. The functional having a sense of propellant consumption is minimized. The analytical solution to the formulated problem is presented. It is shown that the optimal solution can be found in the class of two-impulse control at which the spacecraft’s turn is performed along a free motion trajectory. In order to improve the accuracy of spacecraft guidance into a specified angular position, methods of control are suggested that realize the method of free trajectories. The synthesized controls are invariant with respect to both external perturbations and parametric errors. The results of mathematical modeling are presented that demonstrate high efficiency of developed control algorithms. Propellant consumption for realizing a programmed turn is numerically estimated taking into account considerable gravitational and aerodynamic moments acting upon the spacecraft.     

Optimization Problem in Control over a Programmed Turn of a Spacecraft

The problem of spacecraft reorientation from its initial angular position into a desired final position within a given time interval with a minimum value of the angular moment is considered and solved analytically in this work. It is shown that the control over the spacecraft reorientation, optimal in this sense, might be defined in the class of a regular precession performed by the spacecraft. The moment of the start of deceleration is determined from the principles of the terminal control by using real kinematic parameters of apparatus motion, which increases significantly the accuracy of reorientation. The results of mathematical modeling are presented, showing a high efficiency of the proposed way of reorientation.     

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.     

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.     

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.     

Control of a spacecraft’s spatial turn with minimum value of the path functional

The problem of optimal (with minimum value of the path functional) control over a spatial reorientation of a spacecraft is considered. Using the quaternion method, an analytical solution to this problem is obtained. For the symmetrical optimality index, the complete solution to the problem of spacecraft reorientation is represented in a closed form. The results of mathematical modeling of the spacecraft motion dynamics are presented, demonstrating the practical efficiency of the developed algorithm of control.     

空间机械臂非完整运动规划的遗传算法研究

带空间机械臂航天器系统在无外力矩作用时，系统相对于总质心的动量矩守恒而变为非完整系统。由于非完整约束的不可积性，非完整系统的运动规划与控制比一般系统要困难得多。现利用非完整特性研究了自由漂浮空间机械臂的三维姿态运动控制问题。首先导出带空间机械臂的航天器三维姿态运动数学模型，并将系统的控制问题转化为无漂移系统的非完整运动规划问题。在运动规划中，根据最优控制原理和优化理论，提出基于遗传算法的最优运动规划数值算法。通过数值仿真，表明该方法对空间机械臂及航天器三维姿态运动的非完整运动规划是有效的。     

Nonlinear 6-DOF control of spacecraft docking with inter-satellite electromagnetic force

Compared to traditional docking systems, spacecraft docking with inter-satellite electromagnetic mechanism has distinct advantages. However, its 6-DOF control problem has not been adequately investigated. From our knowledge, this paper attempts to study the 6-DOF control problem for the first time. Based on the far-field electromagnetic force model and Hill's model, the dynamic model of translational motion is derived; using tracking control strategy, LQR method and estimate of Extended State Observer (ESO), an optimal and robust translational controller is designed to satisfy relative position/velocity requirements of soft docking. Representing the attitude of the docking spacecraft pair by unit quaternion, the attitude dynamic and kinematic models with quaternion expression are derived; using behavior-based coordinated control approach and ESO, a decentralized attitude controller is designed to simultaneously align one spacecraft with its absolute desired attitude and with the other spacecraft of the docking pair, requiring no angular velocity measurement and exhibiting better robust capability. The feasibility and performance of this proposed 6-DOF controller are validated by theoretical deduction and simulation results.     

欠驱动刚性航天器姿态的非完整运动规划粒子群算法

研究欠驱动刚性航天器姿态的非完整运动规划问题。众所周知航天器利用三个动量飞轮可以控制其姿态和任意定位，当其中一轮失效，航天器动力学方程表现为不可控。在系统角动量为零的情况下，系统的姿态控制问题可转化为无漂移系统的运动规划问题。基于粒子群优化技术设计了欠驱动刚性航天器姿态的非完整运动规划算法。通过数值仿真，并和遗传算法进行了比较，结果表明该方法对欠驱动航天器姿态运动规划是有效的。     

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.     

基于机动平台序列图像的空间飞行器定轨技术研究

针对机动观测平台单目光学成像系统的特点,在不能测定目标飞行器位置和速度的前提下,通过对成像系统与空间飞行器空间关系的分析,提出了视平均运动角速度与真平均运动角速度的概念,并构建了关于二者的约束方程,实现了基于测角数据的观测斜距的估计,从而解算出定轨所需的初始状态参数。基于观测斜距估计的轨道确定方法把对空间飞行器的定轨问题,归结为根据图像序列计算目标测角和根据测角数据确定观测斜距,解决了利用空间单目光学成像数据的定轨问题,并以高轨卫星为实例对定轨精度进行了仿真验证。     

Impulsive control for angular momentum management of tumbling spacecraft

We discuss an angular momentum control of a tumbling spacecraft. The proposed control method is to apply an impulse by a space robot arm, to measure and control the relative position and attitude between the target spacecraft, and then to apply another impulse until the rotational motion of the target spacecraft is well damped. A discrete controller is designed using the simplified equations of rotational motion through appropriate coordinate transformation. The stationary response under contact model uncertainty is investigated and stability condition is analytically derived. Numerical simulations are given to validate the proposed approach.     

Transition of Planar Rotational Motion of an Asymmetric Spacecraft into Oscillatory Motion at Its Reentry into the Atmosphere

The motion of a spacecraft with small asymmetry relative to its center of mass is considered. The restoring aerodynamic moment of the spacecraft is described by the Fourier series in terms of the angle of attack with the two first sinusoidal and the first cosinusoidal terms. A solution for the angle of attack in the undisturbed rotational motion is found. The analytical expression is obtained for the integral of action taken along the separatrices that separate the rotational and oscillatory regions of the phase portrait of a system. The transition of the spacecraft's motion from planar rotational to oscillatory is investigated. This transition is caused by a slow variation of moment characteristic coefficients, as well as by the presence of small asymmetry and damping and slow variation of their coefficients. Analytical formulas are obtained for determining the times of transition from rotational to oscillatory motion, as well as for the critical angular velocity of beyond-the-atmosphere rotation. When this critical velocity is exceeded, body rotation proceeds for a long time interval (planar autorotation arises).     

Determination of parameters of attitude motion of a small communication satellite using the data of measurements of current of solar panels

A communication satellite (small spacecraft) injected into a geosynchronous orbit is considered. Flywheel engines are used to control the rotational spacecraft motion. The spacecraft after the emergency situation has passed into a state of uncontrolled rotation. In this case, no direct telemetric information about parameters of its rotational motion was accessible. As a result, the problem arose to determine the rotational satellite motion according to the available indirect information: current taken from the solar panels. Telemetric measurements of solar panel current obtained on the time interval of a few hours were simultaneously processed by the least squares method integrating the equations of rotational satellite motion. We present the results of processing 10 intervals of the measurement data allowing one to determine the real rotational spacecraft motion and to estimate the total angular momentum of flywheel engines.     

Algorithms for the transport spacecraft descending stage lateral motion control

This report deals with the problems of synthesizing algorithms for controlling the attitude manoeuver of a transport spacecraft aimed at injecting the spacecraft into a closed terminal domain of “heading-range” phase coordinates which makes it possible to descend to the landing aerodrome region in accordance with a spiral trajectory tracking pattern. The descent trajectory is controlled by changing the roll angle. The principal distinguishing feature of the suggested method of transport spacecraft lateral motion control resides in guiding the spacecraft to a terminal curve and in providing an automatic transfer from roll control to interacting control of roll angle and angle of attack. The performance of the control algorithm under transient conditions are considered in detail.Algorithms controlling the longitudinal range by varing the magnitude of the roll angle and lateral range by selecting the respective sign of the roll control angle are thereafter synthesized separately. The major problem in designing the angular motion control system of transport spacecraft is the development of a high-rate roll axis turn control algorithm. To ensure high accuracy of lateral manoeuvering of the spacecraft it is expedient to accomplish the spacecraft reorientation in roll in a minimum time. It is therewith necessary to take into account with the sideslip angle limitation associated with the need of complying the design conditions of the spacecraft flowaround and with the spacecraft skin selected temperature conditions. It is expected that the total side slip angle is acceptable for measurement. Within the greater portion of the descent trajectory constant-thrust jet-reaction control engines are employed as actuators. Therefore, together with the high speed of response developed control algorithm provides an adequate efficiency of the system from the viewpoint of fuel consumption. The possibilities offered by the suggested algorithms controlling the lateral motions of the center of masses and around the center of masses during the descent stage and in the course of landing approach manoeuvering are illustrated by an example considering a hypothetical transport spacecraft featuring variable aerodynamics and a low frequency of natural oscillations of the angular motion loop. The suggested algorithms make it possible to fully employ the transport spacecraft maneuverability and to meet the terminal heading and velocity requirements within a wide class of disturbances.     

Study of the mode of angular velocity damping for a spacecraft at non-standard situation

Non-standard situation on a spacecraft (Earth’s satellite) is considered, when there are no measurements of the spacecraft’s angular velocity component relative to one of its body axes. Angular velocity measurements are used in controlling spacecraft’s attitude motion by means of flywheels. The arising problem is to study the operation of standard control algorithms in the absence of some necessary measurements. In this work this problem is solved for the algorithm ensuring the damping of spacecraft’s angular velocity. Such a damping is shown to be possible not for all initial conditions of motion. In the general case one of two possible final modes is realized, each described by stable steady-state solutions of the equations of motion. In one of them, the spacecraft’s angular velocity component relative to the axis, for which the measurements are absent, is nonzero. The estimates of the regions of attraction are obtained for these steady-state solutions by numerical calculations. A simple technique is suggested that allows one to eliminate the initial conditions of the angular velocity damping mode from the attraction region of an undesirable solution. Several realizations of this mode that have taken place are reconstructed. This reconstruction was carried out using approximations of telemetry values of the angular velocity components and the total angular momentum of flywheels, obtained at the non-standard situation, by solutions of the equations of spacecraft’s rotational motion.     

Quaternion regularization and trajectory motion control in celestial mechanics and astrodynamics: II

Problems of regularization in celestial mechanics and astrodynamics are considered, and basic regular quaternion models for celestial mechanics and astrodynamics are presented. It is shown that the effectiveness of analytical studies and numerical solutions to boundary value problems of controlling the trajectory motion of spacecraft can be improved by using quaternion models of astrodynamics. In this second part of the paper, specific singularity-type features (division by zero) are considered. They result from using classical equations in angular variables (particularly in Euler variables) in celestial mechanics and astrodynamics and can be eliminated by using Euler (Rodrigues-Hamilton) parameters and Hamilton quaternions. Basic regular (in the above sense) quaternion models of celestial mechanics and astrodynamics are considered; these include equations of trajectory motion written in nonholonomic, orbital, and ideal moving trihedrals whose rotational motions are described by Euler parameters and quaternions of turn; and quaternion equations of instantaneous orbit orientation of a celestial body (spacecraft). New quaternion regular equations are derived for the perturbed three-dimensional two-body problem (spacecraft trajectory motion). These equations are constructed using ideal rectangular Hansen coordinates and quaternion variables, and they have additional advantages over those known for regular Kustaanheimo-Stiefel equations.