The general solution of the HENON–HEILES problem

The general solution of the Henon–Heiles system is approximated inside a domain of the (x, C) of initial conditions (C is the energy constant). The method applied is that described by Poincaré as ‘the only “crack” permitting penetration into the non-integrable problems’ and involves calculation of a dense set of families of periodic solutions that covers the solution space of the problem. In the case of the Henon–Heiles potential we calculated the families of periodic solutions that re-enter after 1–108 oscillations. The density of the set of such families is defined by a pre-assigned parameter ε (Poincaré parameter), which ascertains that at least one periodic solution is computed and available within a distance ε from any point of the domain (x, C) for which the approximate general solution computed. The approximate general solution presented here corresponds to ε = 0.07. The same solution is further improved by “zooming” into four square sub-domain of (x, C), i.e. by computing sufficient number of families that reduce the density parameter to ε = 0.003. Further zooming to reduce the density parameter, say to ε = 10⁻⁶, or even smaller, although easily performable in both areas occupied by stable as well as unstable solutions, was found unnecessary. The stability of all members of each and all families computed was calculated and presented in this paper for both the large solution domain and for the sub-domains. The correspondence between areas of the approximate general solution occupied by stable periodic solutions and Poincaré sections with well-aligned section points and also correspondence between areas occupied by unstable solutions and Poincaré sections with randomly scattered section points is shown by calculating such sections. All calculations were performed using the Runge-Kutta (R-K) 8th order direct integration method and the large output received, consisting of many thousands of families is saved as “Atlas of the General Solution of the Henon–Heiles Problem,” including their stability and is available at request. It is concluded that approximation of the general solution of this system is straightforward and that the chaotic character of its Poincaré sections imposes no limitations or difficulties.     

Effect of perturbations on the non linear stability of triangular points in the restricted three-body problem with variable mass

The effect of small perturbations ε and ε ^′ in the Coriolis and the centrifugal forces, respectively on the nonlinear stability of the triangular points in the restricted three-body problem with variable mass has been studied. It is found that, in the nonlinear sense, the triangular points are stable for all mass ratios in the range of linear stability except for three mass ratios, which depend upon ε, ε ^′ and β, the constant due to the variation in mass governed by Jeans’ law.     

Artificial halo orbits for low-thrust propulsion spacecraft

We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L ₁ and L ₂ in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points inside L ₁ and beyond L ₂ where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum amplitude of halo orbits about artificial equilibrium points inside L ₁ decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L ₂ in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first increases and then decreases after the thrust acceleration exceeds 0.415 mm/s². Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times and require a reasonably small low-thrust acceleration 0.0593 mm/s².     

On the Distance Function Between Two Keplerian Elliptic Orbits

The problem of finding critical points of the distance function between two Keplerian elliptic orbits is reduced to the determination of all real roots of a trigonometric polynomial of degree 8. The coefficients of the polynomial are rational functions of orbital parameters. Using computer algebra methods we show that a polynomial of a smaller degree with such properties does not exist. This fact shows that our result cannot be improved and it allows us to construct an optimal algorithm to find the minimal distance between two Keplerian orbits. This revised version was published online in July 2006 with corrections to the Cover Date.     

利用控制点三维信息标定机载双天线干涉SAR参数

采用基于敏感度方程的方法,研究基于三维重建模型下的机载双天线干涉SAR系统的干涉参数定标问题.干涉参数定标是生成高精度数字高程模型的关键.本文修正了Madsen提出的干涉SAR三维重建的视向量正交分解算法,采用电磁波波前的球面波模型,加入了载机的姿态旋转,构建一种新的干涉SAR三维重建模型.利用各干涉参数对控制点三维信息的不同的敏感性,提出分别利用地面控制点三维信息,对各干涉参数进行定标.并利用中国科学院电子学研究所自主设计、研制的机载干涉SAR系统数据,进行定标处理实验验证.     

一种曲线拟合圆心坐标算法及精度分析

针对在工程应用中,经常需要根据某些圆形物体上的离散点求取其圆心坐标,而采用何种算法简单易行,其精度又与哪些因素有关的相关研究较少的问题,该文提出一种利用曲线拟合求取圆心坐标的简洁算法,并编程实现。然后采用模拟仿真结合统计分析的方法评估了该算法在不同条件下的拟合精度,得出测点误差越大,拟合精度越低;测点个数越多,分布越均匀,拟合精度越高;圆半径对拟合精度影响甚小等结论。在某工程项目中使用该算法得到的结果精度较高,验证了其有效可行,且与采用其他方法得到的坐标值较为接近,说明了该算法的可靠性高。     

一种局部稀疏地面点云与已有DEM的融合方法

机载LiDAR采集的点云数据中会存在一些局部区域地面点稀疏的情况,利用这些稀疏地面点构建DEM时会出现“三角面片化”的问题,严重影响DEM的质量。为此,本文提出了一种局部稀疏地面点云与已有DEM的融合方法：将稀疏点云作为高精度控制点,在尽量保持原始DEM的地形形态特征的前提下,通过高斯核函数加权迭代插值算法对DEM进行高程局部改正,实现稀疏点云与DEM的一致性融合。试验分析表明,融合后的点云数据得到了较好的补充,由此构建的DEM地形形态自然,在精度上相对于融合前的稀疏地面点云有一定改善,在弱精度区域的可靠性有显著提升。     

盾构自动引导系统两棱镜模型的开发与研究

提出2个棱镜加上倾斜仪的盾构自动引导系统模型,利用双轴倾斜仪读取俯仰角和扭转角。根据盾首盾尾以及2个棱镜的空间位置关系,用空间解析几何方法推导了盾构姿态解算方法。