单位矢量法的数学模型(MMUVM)及其简化形式(PUVM1)的迭代法的收敛性  

The Mathematical Model of Unit Vector Method and the Iterative Method for its Simplification

作　　者：陈务深[1] 陆本魁[1] 马静远[1] 

机构地区：[1]中国科学院紫金山天文台

出　　处：《天文学报》2007年第3期343-354,共12页Acta Astronomica Sinica

摘　　要：利用人造地球卫星观测资料测定有摄初轨的单位矢量法(PUVM1),已得到了非常广泛地实际应用.为了对单位矢量法作进一步地完善和改进,首先在考虑测量误差模型的基础上,建立单位矢量法所对应的数学模型MMUVM.它本质上就是一个非线性最优化问题.针对MMUVM,先分别使用多圈仿真数据和实测数据,形成了与之相对应的目标函数,再利用求解最优化问题的一种三对角二次插值模型的直接搜索方法,分别对其进行了数值处理.计算结果表明,所建立的优化模型MMUVM正确合理,所采用的直接搜索方法实用有效.其次,进一步指明了PUVM1和MMUVM之间关系,即:从本质上讲,PUVM1就是MMUVM的一种简化形式.从数学原理上,清楚地解释了利用PUVM1的准法化方程,只能使用单圈短弧段数据进行初始轨道确定,而不能使用长弧段多圈资料进行轨道确定或轨道改进的根本原因.最后,对PUVM1的迭代算法的收敛性问题进行了初步的理论分析,并给出了相应的数值验证实例,指出了PUVM1的迭代格式是条件收敛的,即:只有在满足一定条件后,才能收敛.这也就意味着:有的时候,尽管准法化方程是合情合理的,但是,此时该迭代法却是发散的,无法迭代求出所要的解.In practical application, the perturbed unit vector method （PUVM1） is widely used to determine the orbit by the observation data related to artificial satellites. Some improvement efforts have been done to this method in this paper. Firstly, based on the model of measurement error and the idea of unit vector method, a new mathematical model MMUVM which is essentially a nonlinear optimization problem is introduced. Using simulation data and observation data, the objective function of MMUVM can be formed and solved respectively. Numerical results show that the mathematical model MMUVM is correct and the direct search method, i.e. the quadratic tri-diagonal interpolation DFO method is practical and effective. Secondly, based on the relationship of the semi-normal equation of PUVM1 and the gradient of MMUVM, it points out that MMUVM is a reasonable generalization of PUVM1 and PUVM1 is essentially a simplification to MMUVM. It is a big residual problem when MMUVM is formed by multi-circle long arc data, on the contrary, it is a little residual one when MMUVM is obtained by short arc data. The optimization problem of MMUVM can be solved by a suitable algorithm. The original PUVM1 algorithm is only suitable for the little residual problem. It can be obtained that PUVM1 may be used to determine the preliminary orbit by short arc data, but can not be used to deal with the job of orbit improvement by multi-circle long arc data. Finally, a discussion is made on the convergence of the iterative scheme which is used to solve the semi-normal equation of PUVM1. The corresponding numerical examples are also presented to point out that the iterative scheme of PUVM1 is convergent conditionally. It means that even if there is a reasonable solution to the semi-normal equation of PUVM1, it may be also divergent using the iterative scheme of PUVM1 in some cases.

关 键 词：天体力学 轨道计算和定轨 天体力学 数值方法 

分 类 号：P135[天文地球—天体力学]