Hori method for generalized canonical systems

In this paper, some special features on the canonical version of Hori method, when it is applied to generalized canonical systems (systems of differential equations described by a Hamiltonian function linear in the momenta), are presented. Two different procedures, based on a new approach for the integration theory recently presented for the canonical version, are proposed for determining the new Hamiltonian and the generating function for systems whose differential equations for the coordinates describe a periodic system with one fast phase. These procedures are equivalent and they are directly related to the canonical transformations defined by the general solution of the integrable kernel of the Hamiltonian. They provide the same near-identity transformation for the coordinates obtained through the non-canonical version of Hori method. It is also shown that these procedures are connected to the classic averaging principle through a canonical transformation. As examples, asymptotic solutions of a non-linear oscillations problem and of the elliptic perturbed problem are discussed.     

Long-term evolution of Galileo operational orbits by canonical perturbation theory

Galileo operational orbits are slightly affected by the 3 to 5 tesseral resonance, an effect that can be much more important in the case of disposal orbits. Proceeding by canonical perturbation theory we show that the part of the long-term Hamiltonian corresponding to the non-centralities of the Earth's gravitational potential can be replaced by an intermediary that shows the pendulum dynamics of the 3 to 5 tesseral resonance problem. Inclusion of lunisolar perturbations requires a semi-analytical integration, which is compared with the corresponding results from the well-established Draper Semi-analytical Satellite Theory.     

J2摄动作用下近地轨道卫星编队构形长期演化机理分析

研究近地轨道卫星编队构形在引力摄动作用下的长期演化机理。把用转移矩阵法得出的相对运动分析解表达成一般轨道根数形式，为研究摄动影响而进一步表达成无奇点轨道根数形式；根据各轨道根数的J2摄动解分别研究编队构形在轨道坐标系三个方向上的振幅与相位的长期变化情况，揭示了构形引力摄动的长期演化机理；给出编队构形的摄动表达式，并以仿真实例验证其可信性；最后给出构形长期引力摄动的相关结论，并对编队构形设计提出建议，这将为研究编队构形的规划技术带来方便。     

Luni-solar perturbations in the extended phase space representation of the Vinti problem

Perturbation theory is applied to the Vinti problem—motion about an oblate spheroid—to include the gravitational effects of the sun and moon. The problem is formulated using the extended phase space method which introduces a new independent variable similar to the true anomaly. The disturbing Hamiltonian H₁ for third bodies is of order J₂² (second order) and the final goal is a theory including second order short and long period terms and third order secular terms. The current paper however carries the development only to the second order in the secular terms and the first order in the periodic terms. Problems of including the higher orders are discussed. Therefore, in the development of H₁ all terms of order 10^?9 or larger are retained. The lunar emphemeris retains terms to e′² in the lunar eccentricity. The perturbation analysis is carried out by means of Lie series and is developed through the first order only which is consistent with the final accuracy desired. The generating function W₁ is obtained and separated into the long period, short period and secular terms. From W₁ the coordinates are defined from the Lie series by means of a transformation equation. These coordinates are non-singular for small eccentricity and inclination. Because of the complexity of the equations all algebraic computations were accomplished by means of a computerized Poisson series manipulator developed at the Naval Research Laboratory.     

Dynamics and chaos control of asymmetric gyrostat satellites

The motion of a free gyrostat consisting of a platform with a triaxial ellipsoid of inertia and a rotor with a slight asymmetry with respect to the axis of rotation is considered. Dimensionless equations of motion for a system with perturbations caused by the small asymmetries of the rotor are written in Andoyer-Deprit variables. These perturbations result in a chaotic layer in the separatrix vicinity. Heteroclinic and homoclinic trajectories are written in analytical form for gyrostats with different ratios of their moments of inertia. These trajectories are used to construct a modified Melnikov function, and to produce control that eliminates separatrix chaos. The Poincare sections and Melnikov function are constructed via numerical modeling that demonstrates the effectiveness of control.     

Perturbation effect of the Coulomb drag on the orbital elements of the earth satellite moving in the ionosphere

The perturbation effects of the Coulomb drag on the orbital elements of the earth satellite moving in the ionosphere are studied. The theoretical results show that the Coulomb drag results in both the secular and periodic variation in the semi-major axis and eccentricity. However, the argument of the perigee exhibits no secular variation, but only periodic variation. The inclination and the ascending node remain no variation. As an example, the secular effects of the Coulomb drag on the semi-major axis and the eccentricity of an ionosphere satellite Alouette (S-27) are calculated in the ionosphere with the mean height 1000 km. It can be shown that the semi-major axis contracts and the eccentricity decreases for the case of the Coulomb drag under the interaction of the ions with the electric field of an earth satellite.     

Evolution of asteroid orbits at the 3 : 1 their mean motion resonance with Jupiter (planar problem)

The 3 : 1 mean motion resonance is studied in the planar elliptic restricted three-body problem (Sun-Jupiter-asteroid). Using double numerical averaging, the equations are constructed that describe the secular evolution of eccentricity and perihelion longitude of the asteroid orbit. The region of adiabatic chaos is isolated in the phase space of the system under study.     

The motion of a synchronous satellite in a system similar to the earth-moon system

The stationary motions of a synchronous axisymmetric satellite are studied in the field of attraction by the Earth and a third body whose parameters are close to those of the Moon. Equations of motion are written in canonical variables that take into account the resonance character of the problem. The plots characterizing the dependence of the rotation parameters of the satellite relative to the center of mass on the elements of satellite’s translational motion are presented. A picture is given that represents the initial configuration of the system for implementing stationary motions.     

Pose estimation using linearized rotations and quaternion algebra

In this paper we revisit the topic of how to formulate error terms for estimation problems that involve rotational state variables. We present a first-principles linearization approach that yields multiplicative error terms for unit-length quaternion representations of rotations, as well as for canonical rotation matrices. Quaternion algebra is employed throughout our derivations. We show the utility of our approach through two examples: (i) linearizing a sun sensor measurement error term, and (ii) weighted-least-squares point-cloud alignment.     

交会对接仿真实验方案的研究

提出一种地面交会对接仿真实验方案。其中，由ＣＣＤ相机、三轴位置及三轴姿态转台系统模拟追踪器；由三轴转台及五点光源系统模拟目标器。通过ＣＣＤ相机获取目标五点光源的视频信息，并由视频处理机和计算机进行快速处理。追踪能快速算出目标器的位置和姿态，然后通过伺服系统和执行机构准确跟踪目标。建立了交会对接过程的数学模型，通过仿真验证了模型的有效性。     

Optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field

A complete first-order analytical solution is developed for the problem of optimum low-thrust limited power transfers between neighbouring elliptic non-equatorial orbits in a non-central gravity field. The optimization problem is formulated as a Mayer problem of optimal control with Cartesian elements as state variables. After applying the Pontryagin maximum principle and determining the optimal thrust acceleration, an intrinsic canonical transformation is performed: the Cartesian elements are changed by suitable orbital elements. Hori's method is applied in determining a first-order analytical solution. Simple analytical solutions are obtained explicitly for long-time transfers.     

Elimination of relative secular drift of orbits of two satellites caused by polar oblateness of the Earth

A method of elimination of relative secular drifts in satellite formations is suggested for the case of influence of a perturbation due to polar oblateness of the Earth. The method is applied to eliminate relative secular drifts in the case when a satellite is controlled using an engine mounted along its orientation axis (the satellite is supplied with a passive magnetic attitude control system) and with the help of a solar sail installed on one of the satellites. Analytical results are confirmed by numerical simulation.     

轨道机动的时间-能量综合最优控制

围绕航天器快速精确轨道机动问题,探讨在持续小推力作用下,航天器轨道机动中时 间和能量综合最优控制的技术和方法。基于Pontryagin最小(大)值原理,针对目标轨道为平 面和空间椭圆的情况,推导了时间-能量综合最优控制的Hamilton正则方程组、终端条件 、横截条件和最优控制的表达式,应用数值方法求解正则微分方程组的两点边值问题,得到 了最优控制的数值解,包括最小时间、最小能量、最优轨道、最优控制时变曲线和最优反馈 控制曲线等,实现轨道机动最优控制的精确数值模拟。从数值结果的对比分析中得出了一些 有意义的结论,可供工程实际参考。
     

Evolution of Motion of a Binary Planet

A model of a binary planet, consisting of a material point of small mass and a deformable viscoelastic sphere, is suggested. The center of mass of the binary planet moves in the gravitational field of a central body in the plane, which contains planets forming the binary planet. A deformable spherical planet rotates around the axis orthogonal to the plane of planetary motion. Planet deformations are described by the linear theory of viscoelasticity. It is shown that with an appropriate approximation of the gravitational potential, there is a class of quasicircular orbits, when the eccentricities of an orbit of the center of mass of a binary planet and an orbit, describing mutual planet motion, are equal to zero. The further evolution of motion is investigated in this class of orbits with the use of the canonical Poincare–Andoyer variables. Corresponding averaged equations are found, and phase pictures are constructed; the stability of stationary solutions is investigated on the basis of equations in variations. For the Solar system planets with their satellites, forming binary planets, the application of the presented model allows us to conclude that satellites sooner or later will fall on the corresponding planets.     

Regularization of the Perturbed Spatial Restricted Three-Body Problem by <Emphasis Type="Italic">L</Emphasis>-Transformations

Equations of motion for the perturbed circular restricted three-body problem have been regularized in canonical variables in a moving coordinate system. Two different L-matrices of the fourth order are used in the regularization. Conditions for generalized symplecticity of the constructed transform have been checked. In the unperturbed case, the regular equations have a polynomial structure. The regular equations have been numerically integrated using the Runge–Kutta–Fehlberg method. The results of numerical experiments are given for the Earth–Moon system parameters taking into account the perturbation of the Sun for different L-matrices.     

Stability of large damped flexible spacecraft with stored angular momentum

Vibrational stability of large flexible structurally damped spacecraft carrying internal angular momentum and undergoing large rigid body rotations is analysed modeling the systems as elastic continua. Initially, analytical solutions to the motion of rigid gyrostats under torque-free conditions are developed. The solutions to the gyrostats modeled as axisymmetric and triaxial spacecraft carrying three and two constant speed momentum wheels, respectively, with spin axes aligned with body principal axes are shown to be complicated. These represent extensions of solutions for simpler cases existing in the literature. Using these solutions and modal analysis, the vibrational equations are reduced to linear ordinary differential equations. Equations with periodically varying coefficients are analysed applying Floquet theory. Study of a few typical beam- and plate-like spacecraft configurations indicate that the introduction of a single reaction wheel into an axisymmetric satellite does not alter the stability criterion. However, introduction of constant speed rotors deteriorates vibrational stability. Effects of structural damping and vehicle inertia ratio are also studied.     

Evolution of Rotational Motion of a Symmetrical Satellite with Viscoelastic Rods

We study the motion of a symmetrical satellite with a pair of flexible viscoelastic rods in a central Newtonian gravitational field. A restricted problem formulation is considered, when the satellite's center of mass moves along a fixed circular orbit. A small parameter is introduced which is inversely proportional to the stiffness of flexible elements. Another small parameter is equal to the ratio of the squared orbital angular velocity and the squared magnitude of the initial angular velocity of the satellite. In order to describe the satellite rotational motion relative to the center of mass, we use the canonical Andoyer variables. In the undisturbed formulation of the problem, i.e., at = 0 and = 0, these variables are the action–angle variables. Equations describing the evolution of motion are derived by an asymptotic method which combines the method of separating motions for systems with an infinite number of degrees of freedom and the Krylov–Bogolyubov method for systems with fast and slow variables. The manifolds of stationary motions are found, and their stability is investigated on the basis of equations in variations. Phase portraits are constructed which describe the rotational motion of a satellite at the stage of slow dissipative evolution.     

惯导系统可观测性及最佳可观测子空间的定量研究

应用结构分解理论及奇异值分解法对惯导系统（INS）误差模型的可观测性进行了全面分析。研究了一种基于可观测性定义及奇异值分解法结合的定量确定惯导系统可观测性及可观测度的方法，明确了惯导系统的三个最佳不可观测状态组合；应用线性控制系统的Kalman典范结构分解定理，确定了惯导系统的最佳观测子系统。此项研究成果，为进一步研究INS的快速精确对准及标定技术奠定了理论基础。     

Maximum Lyapunov Exponents for Chaotic Rotation of Natural Planetary Satellites

Based on the Chirikov approach [1, 2] within the context of the theory of separatrix mappings, we suggest and substantiate a simple method for estimating the maximum Lyapunov characteristic exponent (MLCE) of motion in a chaotic layer in the neighborhood of nonlinear resonance separatrix of a Hamiltonian system under an asymmetric periodic perturbation. For a number of natural planetary satellites, using this method, the estimates are made of the MLCE of chaotic rotation (relative to the center of mass) in the main chaotic layer, i.e., in the chaotic layer in the neighborhood of a synchronous resonance separatrix. The value of the MLCE is determined by an orbital eccentricity and by the parameter of the satellite's dynamic asymmetry. The quantity inverse to the MLCE gives a typical time of predictable rotational motion.