A global view of the isochronous trajectory families by Hamilton's hodographic transformation

An overall view of the characteristic features of the isochronous trajectory families is presented. The trajectories are assumed Keplerian, isoenergetic, and satisfying Lambert's conditions, that is the sum of radial distances of the two termini, and their linear separation are both constant. A geometric approach is adopted by using Hamilton's hodographic transformation. The connection between Hamilton's hodographic isochronism and the Lambert's theorem is briefly reviewed, and the interrelations among the Hamiltonian set of parameters and the Lambertian sets are developed. The totality of the isochronous trajectory families is then viewed in the velocity space. Various Lambertian invariants as well as some essential noninvariants are examined, and relevant formulas derived. The salient features of some particular isochronous families are then summarized; and Hamilton's time integral is briefly treated. The analysis is essentially two-dimensional, with its extension to the three-dimensional case finally discussed.     

A remark concerning fundamental rocket theory

This Note attempts to clear up the discussions relating to the interpretation of Tsiolkowskiy's equation vs Irving's solution.     

多饱和约束拦截弹参数时变系统控制器设计

针对拦截弹参数时变和执行机构饱和非线性影响控制系统性能,提出基于齐次多项式参数相关 Lyapunov 理论和非线性矩阵微分方程凸多面体转换算法相结合的鲁棒最优控制系统设计方案。 通过引入标准饱和约束算子,建立控制系统数学模型并给出控制系统结构;引入饱和归一化因子,把非线性参数时变微分方程转化成线性多胞型微分方程;基于齐次多项式参数相关Lyapunov理论和拓展Pólya’s定理,把参数时变多胞型微分方程转换成齐次多项式参数相关微分方程,并给出齐次多项式参数相关控制器结构;利用迭代算法把非线性矩阵不等式转换成线性矩阵不等式(LMIs),进而基于LMI凸优化理论获得控制器解,设计出的控制器是关于齐次多项式阶次g和Pólya’s松弛度d的函数;通过定点和制导控制系统仿真校验了该算法,仿真结果表明随着齐次多项式阶次g和Pólya’s松弛度d的增加,设计出的控制器在确保系统稳定的前提下改善控制系统性能。     

Design of phase-locked loop for communication system via on-board signal processing satellites

The energy inconsistency for a multiple access system via an active satellite with simple repeating of signals is claimed. The on-board ideal signal processing algorithm resolving this inconsistency is considered and the optimal group signal processing algorithm in a manner of digital signals shape transformation followed by their time compression is suggested. The necessity for individual clock synchronization of ground stations is shown and the synchronous multiple access system block diagram with an optimal on-board signal processing is discussed.A phase disturbance dynamics of a two-loops clock synchronization system with variable time delay is investigated. A nonlinear differential-difference equation is obtained and for the first order PLL with time delay the exact solutions are found in cases of relay and linear shapes phase discriminator's characteristics. Phase errors' expressions are derived and the main factors of accuracy limiting caused by PLL's feedback time delay are established.     

Optimal transfer from the lagrangian points

The transfer from the equilateral Lagrangian points of the Earth-Moon system is analysed. The final states of the velocity of the space vehicles and of the rotation velocity of the propulsion vector are assumed given. The trajectory which ensures the transfer in optimal time consists of three arcs. On this trajectory the rotation velocity of the direction of the propulsion has the extremal value or corresponds to the Lawden's tangent law. The use of the matching of the arcs together with transversality conditions and final conditions determines the constants of integration and the evolution time. The resulting parametric equations of the optimal trajectory are of integral form.     

Stability of large flexible damped spacecraft modeled as elastic continua

Vibrational stability of a large flexible, structurally damped spacecraft subject to large rigid body rotations is analysed modelling the system as an elastic continuum. Using solution of rigid body attitude motion under torque free conditions and modal analysis, the vibrational equations are reduced to ordinary differential equations with time-varying coefficients. Stability analysis is carried out using Floquet theory and Sonin-Polya theorem. The cases of spinning and non-spinning spacecraft idealized as a flexible beam plate undergoing simple structural vibration are analysed in detail. The critical damping required for stabilization is shown to be a function of the spacecraft's inertia ratio and the level of disturbance.     

视场约束的交会对接V-bar撤离控制研究

利用解析和数值结合的方法研究了在视场约束条件下交会对接V-bar撤离的控制问题。描述了交会对接V-bar撤离的径向冲量机动方案，说明了径向机动撤离方案具有减少羽流污染和故障情况下能够避免飞行器碰撞的优点。基于CW方程推导出了机动点位于V-bar上的特殊情况下径向冲量和视场角的解析公式，同时也得到了求最大视场角时刻和冲量作用后V-bar方向撤离距离的计算公式。对于任意机动点的一般情况，得到了求解径向冲量和视场角关系的非线性方程，这一方程可利用数值方法求解。最后，通过数值仿真验证了该方法的正确性。     

RLV末端能量管理段轨迹在线规划与制导

针对可重复使用飞行器（RLV）末端能量管理段利用数值优化算法在线规划轨迹的实时性无法保证,工程应用性差的问题,在末端能量管理段的航向调整段（HAC）轨迹前增加直线预测捕获段(PASL),并在该阶段提前完成末端区域能量管理(TAEM)段轨迹的在线规划,从而降低工程中对轨迹在线规划方法实时性的高要求。首先,通过求解零气动角下的质点运动方程解析解,得到直线预测捕获段结束点的飞行状态预测值作为TAEM段轨迹的初始点状态。然后,在线求解以航向调整段进入点飞行器航向角偏差最小为目标函数,以动压、过载和速度滚转角限制为约束的非线性规划问题,得到航向调整螺旋线中心的最优位置。最后,设计了以规划轨迹确定的标称气动角指令为前馈,以跟踪偏差的比例+微分律生成指令(PD)为反馈的TAEM段制导律。算例仿真校验了本文基于弹道预测的末端能量管理方法的有效性。     

Trajectory correction maneuver design using an improved B-plane targeting method

A spacecraft for interplanetary mission is usually perturbed by some disturbance sources. The trajectory correction maneuver (TCM) is required to adjust this trajectory error, and the B-plane targeting method is widely used in this field. However, this B-plane targeting method is based on the differential correction algorithm, and a numerical Jacobian matrix is usually used for this algorithm. Therefore, our main goal in this paper is to suggest the improved B-plane targeting method to overcome the disadvantages of the conventional B-plane method which requires a numerical Jacobian matrix for the initial perturbation selection and iterations. For this improvement, an analytical Jacobian matrix is introduced instead of the numerical Jacobian matrix. Then, another B-plane approach that offers an analytical solution is suggested using the target eccentricity instead of the target time of closest approach (TCA). Using a modified Kepler's equation, the previous B-plane targeting approach can be replaced with the new method through the analytical solution.     

基于正则摄动理论的无动力飞行器末段制导方法

以无动力飞行器末制导问题为研究对象,用正则摄动理论对其纵向弹道方程进行处理。在零阶方程中考虑重力和气动力的主导部分,在一阶方程中考虑重力和气动力的剩余部分,进而获得了弹道方程的解析解,仿真证明了该解析解具有较高的计算精度和计算效率。基于弹道解析解所获得的脱靶量,提出了正则摄动制导方法,仿真证明了该方法具有最优性。     

强稳定的最优单步法公式

本文研究一阶常微分方程灵敏值解的收敛性与稳定性，利用最优化方法，确定最优系数，导出两个强稳定的单步公式，并加以优化和改进，得到新的算法。经过实际计算，结果优目前的单步公式。同时也验证了梯形公式。     

一种燃料最省的火星精确着陆动力下降段快速轨迹优化方法

针对火星精确着陆问题,给出一种燃料最省的火星着陆动力下降段快速轨迹优化方法。首先以燃料最省为指标建立了着陆器动力下降段轨迹优化问题模型,然后利用极大值原理对该模型的推力大小、推力方向与协状态变量之间的关系进行分析,得出着陆器推力只能以最大或最小推力工作、推力方向和与速度对应的协状态矢量方向相同的结论。并以此为基础,将原问题的状态微分方程和协状态微分方程转换为有限个推力为常值的分段函数,推导每段内微分方程的解析表达式,给出了动力下降段轨迹优化方法的算法流程;最后通过数学仿真将所提出的方法与凸规划、多项式制导方法进行比较。结果表明该方法不仅避免了多项式方法没有考虑推力约束及燃料消耗的缺陷,而且在计算效率方面较凸规划方法有大幅度提高。     

Optimization of low-thrust orbit transfers with constraints

Approximate numerical methods of optimization are presented for multi-orbit noncoplanar orbit transfers of low-thrust spacecraft. The linear representation of derivatives of boundary parameters is used in the vicinity of a reference trajectory with its discretization into small segments. For each segment a set of pseudo-impulses is introduced, representing possible directions of the thrust vector. In order to solve essentially nonlinear problems, the iterative process of upgrading the reference trajectory is used. At each iteration the linear programming problem of high dimensionality is solved, its boundary conditions being represented in the form of a linear matrix equation. Interval constraints are considered in the form of blocking the propulsion system operation on shadow segments of the orbit, as well as cycling constraints, and constraints on total duration of maneuvers at certain trajectory segments. The results of comparison with solutions obtained by other methods are presented together with examples illustrating the convergence of iterative processes. Optimizations of the trajectories for launching geosynchronous satellites to their orbits and of the trajectories of a noncoplanar transfer from low to high-elliptic Molniya orbit are considered under these constraints.     

The darboux point and the conjugate point on optimal deorbit for reentry trajectories

The concept of the Darboux point at which an extremal loses its global optimality is extended to the case of discontinuous control. Using Contensou's domain of maneuverability, the condition for optimal switching at a corner is derived and the optimality of the trajectory in the neighborhood of a Darboux point is analyzed. The theory is applied to the problems of minimum-fuel planar and noncoplanar deorbit from elliptical orbits for atmospheric entry at a prescribed angle. In each case, the global optimal trajectory is assessed and it is found that in these nonlinear problems the Darboux point and the conjugate point are distinct. The global optimality is always lost before local optimality.     

On autonomous tracking of a minor celestial body

There is suggested an algorithm of the autonomous tracking of a minor celestial body (comet, asteroid or satellite) by the photo-TV-camera of the spacecraft during its flyby in the vicinity of the celestial body, investigated by the data of images obtained by this camera. This algorithm allows prediction of the celestial body angular position in order to point the onboard camera to the body. The field of applicability for the suggested algorithm taking into account various errors is being investigated. The problems of the optimization of the program of the autonomous tracking in order to decrease the prediction errors or to increase the prediction time interval are considered. The results obtained are illustrated by the example of the autonomous tracking of Halley's comet nucleus.     

高度-速度剖面内再入轨迹快速规划

提出一种升力式再入航天器进入稠密大气后的轨迹规划方法。在预先设定攻角剖面的前提下,利用路径约束(驻点热流、动压和过载)在高度-速度(H-V)剖面内直接获得轨迹下边界;利用终端约束确定以轨迹下边界为基准的高度增量,进而通过下边界与高度增量的加和形成满足要求的再入轨迹。其中,增量的形式选取为分段二次型函数,其大小可通过割线法快速获得。倾斜角大小可根据纵向动力学方程反解得到,其方向依据航向误差角走廊确定。通过对典型工况的仿真,结果表明所提方法能够快速规划出再入轨迹,且适应性好。     

吸气式高速飞行器爬升-巡航轨迹在线优化

针对吸气式高速飞行器的大气层内爬升和巡航飞行段,进行了在线轨迹优化设计。与传统的爬升段结合定高定速巡航轨迹形式不同,采用优化的方法得到的轨迹能够保证性能指标的最优性。首先,将轨迹优化问题建模为非线性最优控制问题,控制量包括攻角、倾侧角以及燃油当量比。然后,对问题的非线性进行了处理,将变量进行离散化,得到凸优化问题。分析了飞行器的气动和推力特性,设计了优化变量的初始参考。最后,设计算法的外环迭代策略,将每一次求得的解作为参考轨迹进行下一次迭代计算。数值仿真结果表明,该方法能够解决吸气式高速飞行器爬升巡航段轨迹优化问题,并且求解时间满足在线轨迹优化要求。     

远程星际航行补篇

通过一系列变量置换推导出不同喷气速度和不同加速度条件下,计算恒星际飞船加速段的时间和距离的普遍公式及多级火箭加速情况下加速段的时间和距离的公式。最后以天狼星作为航行目标、计算了用5级和10级火箭加速飞行的时间和距离。结果表明,当火箭的级数很多,且各级加速度相等时,其飞行时间与距离非常接近匀加速飞行的情况。     

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.     

水平着陆飞行器着陆架系统耦合压缩触地过程仿真分析

水平着陆飞行器着陆架触地是一个短时间、高动态的复杂运动过程,着陆架系统中的轮胎和减震器压缩过程存在耦合因素,直接求解其微分方程具有一定的困难。采用一种基于力矩瞬时平衡条件的触地过程仿真分析方法,通过求解以减震器压缩量增量为自变量的一元非线性方程,得到既满足力矩瞬时平衡条件又满足几何耦合关系的减震器压缩量和轮胎压缩量。该方法不需要求解着陆架系统的运动方程,可在飞行器六自由度仿真模型基础上直接扩展,无需改变其仿真步长,可有效应用于水平着陆飞行器着陆架系统耦合压缩触地过程的仿真分析。