基于自动微分方法的SGP4模型的雅可比矩阵计算

Automatic Differentiation for the Computation of Jacobian Matrix for SGP4 Model

美国双行根数（TLE）是北美防空司令部提供的空间目标和碎片轨道数据。自1980年著名的SpaceTrack Re．portNo3．公开了其配套的外推模型SGP4和程序后，TLE成为了空间目标和碎片观测与研究最主要的数据源。随着设备成本的降低，越来越多的国家、组织甚至个人都能够开展空间目标与碎片观测，利用独立数据定轨生成TLE兼容的轨道根数无疑是最为有利的方式之一，特别是可提供原始数据中没有的协方差信息。雅可比矩阵的计算是SGP4模型定轨过程中最为复杂的环节，过去研究采用了有限差分方法进行计算。近年兴起了一种新的微分求解方法——自动微分方法，为雅可比矩阵求解提供了一种新的思路。本文将公开的SGP4模型外推程序代码修改为支持自动微分的代码，并计算了雅可比矩阵。试验表明相对于有限差分方法，自动微分方法在多方面有明显优势，可在相关软件中推广使用。

Two Line Element is the orbital data source supplied by NORAD. Since the publication of famous Space Track Report No. 3 in 1980, which gave out the algorithm of corresponding SGP series models for propagation with TLE, TLEs became the most widely used data in the observation and research of space object and debris. With the decrease in the cost of instrument building, more and more countries, organization and even individual person can make the measurements of the space objects. With these independent data, to generate TLE compatible result with orbit determination is beneficial for supplying convariance information of which original da- ta are lack. The computation of Jacobi matrix is the most complex part of orbit determination with SGP4 model, which was implemented with finite difference method in past research. In recent years, a state-of-the-art technique called automatic differentiation becomes popular and gives a novel approach for the calculation of Jacobian matrix. In this paper, the open source code of SGP4 propagation is adaped to support automatic differentiation and to give the Jacobian matrix in a concise way. Experiment shows that new method has advantage in many aspects, and can be employed in related software coding.