Quaternion regularization in celestial mechanics and astrodynamics and trajectory motion control. I

Regularization problems in celestial mechanics and astrodynamics are considered. The fundamental regular quaternion models of celestial mechanics and astrodynamics are presented. It is shown that the efficiency of analytical investigation and numerical solution of boundary problems of optimal trajectory motion control of spacecraft may be increased using quaternion astrodynamics models. The regularization problem of celestial mechanics and astrodynamics that implies eliminating the feature, which arises in the equations of the two-body problem in case of impact of the second body with the central body, is considered in the first section of the paper. The quaternion method for regularizing the equations of the perturbed spatial two-body problem suggested by the author is presented; the method is compared with Kustaanheimo-Stiefel (KS) regularization. Demonstrative geometric and kinematic interpretations of regularizing transformations are provided. Regular quaternion equations for the two-body problem, which generalize the regular Kustaanheimo-Stiefel equations, as well as regular equations in quaternion osculating elements and quaternion regular equations for perturbed central motion of a material point, are considered. The papers on quaternion regularization in celestial mechanics and astrodynamics are briefly analyzed.     

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.     

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.     

Optimization of passive orbit with the use of gravity maneuver

A high-precision method of calculating gravitational interactions is applied in order to determine optimal trajectories. A number of problems, necessary for determination of optimal parameters at a launch of a spacecraft and during its flyby near celestial bodies, are considered. The spacecraft trajectory was determined by numerical integration of the equations of passive motion of the spacecraft and of the equations of motion for planets, the Sun, and the Moon. The optimal trajectory of the spacecraft approaching the Sun is determined by fitting its initial conditions.     

双欧法在克服伞-弹系统欧拉方程奇异性中的应用

文章针对伞弹系统欧拉方程奇异性 ,利用正、反欧拉方程精华区、奇异区呈倒挂的原理 ,建立了一套与飞机不同的反欧拉系统。将这套双欧系统应用在克服伞 -弹系统欧拉方程奇异性中 ,取得了满意的结果。同时对双欧法与四元数法计算结果进行了对比验证 ,从而更加清楚地看到了四元数法的方法缺陷和不足 ,为进一步能将双欧法应用于伞 -弹系统六自由度弹道实时仿真进行了有益的尝试     

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.     

一种航天器空间机动轨道的改进形状设计方法

针对带推力约束的航天器三维空间机动轨道初始设计问题,提出了一种基于傅立叶级数展开的改进形状设计方法。首先,在柱坐标系下建立了航天器运动模型,并将基于傅立叶级数展开的形状方法推广到三维空间的机动轨道初始设计;然后,基于所得到的空间初始机动轨道,采用直接配点法进行了完整三维空间机动轨道的优化设计。仿真结果表明,提出的方法可以为带有任务约束的航天器三维空间机动轨道的优化设计提供更优的初始参数及其解析解,为空间机动轨道设计提供了新的方法。     

An alternative algorithm for the spacecraft information system with magnetometer and solar sensor

The possibility of identification of motion parameters of a low-orbit spacecraft using readings of a three-axis magnetometer and solar position sensor, without integration of the Euler’s dynamic equations or direct measurement of the object’s angular velocity, is considered.     

连续力矩作用下的柔性航天器再定向与振动抑制

研究带两帆板航天器的三维再定向与振动抑制问题，执行机构为反作用轮。建立了柔性空间飞行器三轴耦合姿态动力学模型和四元数姿态运动学模型，建模时考虑了帆板的弯曲变形和扭转变形。采用拟欧拉角及角速度作为反馈信号，设计了一种ＰＤ控制律。该控制律可以用于对任意目标姿态的再定向而形式保持不变。用Ｌｙａｐｕｎｏｖ方法证明了姿态的渐进稳定性和模态振动和衰减性。非线性闭环系统的仿真结果验证了所设计控制律不仅能够使航天     

Influence of disturbances on the angular motion of a spacecraft in the powered section of its descent

The motion of a variable-mass spacecraft is considered in the powered section of a descending trajectory. Approximate analytical solutions are obtained for the angles of spatial orientation of the spacecraft, which allows one to analyze the nutation motion and to develop recommendations on the spacecraft’s mass configuration, providing the smallest possible deviations of the longitudinal axis and thrust vector from specified directions. The errors of stabilization of the spacecraft’s longitudinal axis are calculated by means of numerical integration of complete models and using the obtained analytical solutions, the results being in good agreement.     

Adaptive sliding mode control for six-DOF relative motion of spacecraft with input constraint

This paper addresses the synchronized control problem of relative position and attitude for spacecraft with input constraint. First, using dual quaternion, the kinematic and dynamic models of the six-degree-of-freedom relative motion of spacecraft are introduced. Second, a new adaptive sliding mode control scheme is proposed to guarantee the globally asymptotic convergence of relative motion despite the presence of control input constraint, parametric uncertainties and external disturbances. A detailed stability analysis of the resulting closed-loop system is included. Finally, simulation results are presented to illustrate the validity and effectiveness of the proposed controller, which has the following properties: (1) explicit accounting for the problem of input constraint, (2) fast convergent rate and accurate results can be obtained, (3) no chattering phenomenon is present in the control torque and control force, (4) self-adaptive regulation law is dynamically adjusted to ensure the tracking errors tend to zero asymptotically, (5) the upper bounds of unknown variables are estimated dynamically.     

Cubature Kalman filtering for relative spacecraft attitude and position estimation

A novel relative spacecraft attitude and position estimation approach based on cubature Kalman filter is derived. The integrated sensor suit comprises the gyro sensors on each spacecraft and a vision-based navigation system on the deputy spacecraft. In the traditional algorithm, an assumption that the chief?s body frame coincides with its Local Vertical Local Horizontal (LVLH) frame is made to construct the line-of-sight observations for convenience. To solve this problem, two relative quaternions that map the chief?s LVLH frame to the deputy and chief body frames are involved. The general relative equations of motion for eccentric orbits are used to describe the positional dynamics. The implementation equations for the cubature Kalman filter are derived. Simulation results indicate that the proposed filter provides more accurate estimates of relative attitude and position over than the extended Kalman filter.     

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.     

A Proposal for Constructing a Group of Space Objects for Solving Transport and Energy Problems

We propose to develop a scheme for a group of space objects which includes a set of orbital tethers and exchange masses. These objects are injected into circumterrestrial orbits. A variety of problems can be solved with the help of this space system, namely, transport problems and problems of converting the electric power generated onboard spacecraft into mechanical energy of motion of the space objects. In the future, natural celestial bodies (in particular, the Moon) can be considered as elements of the system. This opens up the possibility of using the energy of motion of the Moon both for solving transport problems and for generating electric power onboard spacecraft.     

Target berthing and base reorientation of free-floating space robotic system after capturing

Space robots are playing an increasingly important role in on-orbital servicing, including repairing, refueling, or de-orbiting the satellite. The target must be captured and berthed before the servicing task starts. However, the attitude of the base may lean much and needs re-orientating after capturing. In this paper, a method is proposed to berth the target, and re-orientate the base at the same time, using manipulator motion only. Firstly, the system state is formed of the attitude quaternion and joint variables, and the joint paths are parameterized by sinusoidal functions. Then, the trajectory planning is transformed to an optimization problem. The cost function, defined according to the accuracy requirements of system variables, is the function of the parameters to be determined. Finally, we solve the parameters using the particle swarm optimization algorithm. Two typical cases of the spacecraft with a 6-DOF manipulator are dynamically simulated, one is that the variation of base attitude is limited; the other is that both the base attitude and the joint rates are constrained. The simulation results verify the presented method.     

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.     

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.     

Periodic motions of a satellite-gyrostat relative to its center of mass under the action of gravitational torque

We investigated periodic motions of the axis of symmetry of a model satellite of the Earth, which are similar to the motions of the longitudinal axes of the Mir orbital station in 1999–2001 and the Foton-M3 satellite in 2007. The motions of these spacecraft represented weakly disturbed regular Euler precession with the angular momentum vector of motion relative to the center of mass close to the orbital plane. The direction of this vector during the motion was not practically changed. The model satellite represents an axisymmetric gyrostat with gyrostatic moment directed along the axis of symmetry. The satellite moves in a circular orbit and undergoes the action of the gravitational torque. The motion of the axis of symmetry of this satellite relative to the absolute space is described by fourth-order differential equations with periodic coefficients. The periodic solutions to this system with special symmetry properties are constructed using analytical and numerical methods.     

6-DOF robust adaptive terminal sliding mode control for spacecraft formation flying

This paper addresses the tracking control problem of the leader–follower spacecraft formation, by which we mean that the relative motion between the leader and the follower is required to track a desired time-varying trajectory given in advance. Using dual number, the six-degree-of-freedom motion of the follower spacecraft relative to the leader spacecraft is modeled, where the coupling effect between the translational motion and the rotational one is accounted. A robust adaptive terminal sliding mode control law, including the adaptive algorithms, is proposed to ensure the finite time convergence of the relative motion tracking errors despite the presence of model uncertainties and external disturbances, based on which a modified controller is furthermore developed to solve the dual-equilibrium problem caused by dual quaternion representation. In addition, to alleviate the chattering, hyperbolic tangent function is adopted to substitute for the sign function. And by theoretical analysis, it is proved that the tracking error in such case will converge to a neighborhood of the origin in finite time. Finally, numerical simulations are performed to demonstrate the validity of the proposed approaches.