Zhang Yanli, Wang Haoquan, Zheng Jingfeng, Hua Lei, Zhang Rui, Xie Jian
School of Information Innovation and Big Data, Shanxi Jinzhong Institute of Technology, Jinzhong, 030600, China.
School of Information and Communication Engineering, North China University, Taiyuan, 030051, China.
Sci Rep. 2025 Jul 18;15(1):26060. doi: 10.1038/s41598-025-10236-2.
To address the issues of low accuracy and time-difference ambiguity in the localization and tracking of multiple satellites, we have conducted a thorough study of the localization algorithm and the ambiguity resolution algorithm for the time-difference fusion of multiple satellites. The localization algorithm based on Gaussian-Newton iteration for time-difference fusion of three satellites is proposed. The time-difference equation observed during the satellite overhead is combined with the elevation observation equation to construct the cost function for the overdetermined case, and the nonlinear least squares problem is solved based on Gaussian-Newton iteration. For localization and tracking for time-difference fusion, the Kalman filtering combined with Gaussian mixture model (KFGMM) is proposed as the ambiguity resolution algorithm of time-difference for stationary targets; the capacitive Kalman filtering combined with Gaussian mixture model (CKFGMM) is proposed for cruising targets. The mathematical model of time-difference ambiguity is established, the calculation method of time-difference window and number of ambiguous time-difference is given, and the measurements of ambiguous time-difference are approximated by Gaussian mixture model. Experiments show that localization algorithm for time-difference fusion of multiple satellites outperforms other advanced localization methods and achieves the Cramér-Rao Lower Bound (CRLB) of fusion localization; with the increase of filtering time, the ambiguity resolution algorithm for time-difference fusion of multiple satellites can reach the Bayesian Cramér-Rao Lower Bound (BCRLB) and outperforms the algorithm combining with the direction finding assistance for ambiguity resolution.
为了解决多颗卫星定位与跟踪中精度低和时差模糊度的问题，我们对多颗卫星时差融合的定位算法和模糊度解算算法进行了深入研究。提出了基于高斯 - 牛顿迭代的三颗卫星时差融合定位算法。将卫星过境时观测到的时差方程与仰角观测方程相结合，构建超定情况下的代价函数，并基于高斯 - 牛顿迭代求解非线性最小二乘问题。对于时差融合的定位与跟踪，提出了卡尔曼滤波结合高斯混合模型（KFGMM）作为静止目标时差的模糊度解算算法；对于巡航目标，提出了电容卡尔曼滤波结合高斯混合模型（CKFGMM）。建立了时差模糊度的数学模型，给出了时差窗口和模糊时差数量的计算方法，并用高斯混合模型对模糊时差测量值进行近似。实验表明，多颗卫星时差融合定位算法优于其他先进定位方法，达到了融合定位的克拉美 - 罗下界（CRLB）；随着滤波时间的增加，多颗卫星时差融合的模糊度解算算法能够达到贝叶斯克拉美 - 罗下界（BCRLB），并且优于结合测向辅助进行模糊度解算的算法。
