Some issues of time-optimal control over a spacecraft programmed turn

The problem of optimal turn of a spacecraft from an arbitrary initial position to a final specified angular position in a minimum time is considered and solved. A case is investigated, when the constraint on spacecraft’s angular momentum during the turn is essential. Based on the quaternion method a solution to the posed problem has been found, and an optimal control program is constructed taking the constraints on controlling moment into account. The optimal control is found in the class of regular motions. A condition (calculation expression) is presented for determining the moment to begin braking with the use of measurements of current motion parameters, which considerably improves the accuracy of putting the spacecraft into a preset position. For a dynamically symmetrical spacecraft the solution to the problem of optimal control by the spacecraft spatial turn is presented in analytical form (expressions in elementary functions). An example of mathematical modeling of the spacecraft motion dynamics under optimal control over reorientation is given.     

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.     

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。     

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.     

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.     

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。     

零角动量欠驱动航天器逆最优稳定控制律设计

研究轮控式零角动量欠驱动航天器姿态最优稳定控制问题。考虑到该类型航天器不存在定常光滑稳定控制律的特点,通过Lyapunov直接法和Backstepping方法设计了一种非线性不连续反馈控制律,同时得到控制Lyapunov函数（CLF）,并由此得到逆最优稳定控制律。该控制律可以避开求解Hamilton-Jacobi方程,最小化某一代价函数,同时具有扇形稳定裕度,对输入不确定性具有一定的鲁棒性。数学仿真结果表明,所设计的非线性不连续反馈控制律能够使姿态渐近稳定至平衡点,并具有最优性,以及在转动惯量存在不确定性时,扇形稳定裕度使系统具有一定的鲁棒性。     

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.     

航天器星敏感器自主定位方法及精度分析

用星敏感器和地平仪测量星光与地平之间的“星光抑角”为观测量，利用推广卡尔曼滤波方法实时估计航天器的最佳位置，使航天器在失去地面遥控的情况下，能够自主准确地确定运行轨道。由于航天器自主定位系统在工作期间会受到硬、软件等诸多因素的影响，因而使其定位精度达不到预计要求。在此我们通过大量仿真计算，指出一些对自主定位系统精度影响较大的因素，并对它们进行了比较分析。     

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.     

目标捕获后航天器组合体的角动量转移与抑振规划

针对失稳目标捕获后航天器组合体的位姿调整与稳定问题，提出一种组合体角动量转移与振动抑制复合规划方法。首先建立了同时考虑了空间机械臂、目标卫星太阳翼、服务卫星太阳翼等柔性构件的航天器组合体动力学模型。然后提出角动量转移优化方法，规划机械臂最终构型，保证组合体相对稳定后的角速度最小；基于粒子群算法设计了机械臂最优抑振轨迹规划方法，抑制角动量转移过程中的机械臂和太阳翼的柔性振动。最后通过数值仿真验证了规划方法的有效性。仿真结果表明，该方法能够有效实现组合体的角动量转移，并显著降低组合体的柔性振动，具有工程实用性。     

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.     

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.     

Analysis of evolution of multi-orbit perturbed trajectories of a spacecraft in a central field of attraction

Properties of differential equations of multi-orbit trajectory motion of a spacecraft are investigated analytically. The spacecraft moves under the action of small perturbations (in particular, low thrust) in the plane of a central Newtonian field of attraction. The conditions are specified for existence of a partial singular aperiodic solution, in the neighborhood of which the behavior of osculating elements changes sharply. In this case, phase variables (the angular position of the pericenter and the true anomaly) are found to undergo the sharpest changes. The exact superposition of solutions is suggested for the equations of motion transformed to the form of a quasi-linear, weakly non-stationary system: a partial singular aperiodic solution and fast solutions oscillating around it. Asymptotic representations are obtained for both components of the superposition. They are fairly exact in the region of smallness of perturbing terms at a long variation of the argument.     

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。     

欠驱动航天器姿态控制系统的退步控制设计方法

应用退步控制设计方法研究欠驱动航天器的姿态控制问题。将系统分为运动学和动力学两个子系统分别进行控制律的设计。首先导出了一种动力学子系统的镇定控制律，以减低失控轴的角速度分量对运动学子系统的影响。在此基础上，假设这一角速度分量为小量，利用运动学中的角速度交叉耦合项对失控轴的姿态进行控制。通过推导出角速度中间控制律，实现了运动学子系统的镇定，并进一步设计了姿态退步控制律。最后进行了数值仿真，验证了所推导的控制律的有效性。