轨道预报的一种乘法保辛摄动方法
|
摘要
为了准确预报人造卫星的轨道,需要考虑复杂的动力学模型,并使用数值方法进行求解.虽然Runge-Kutta法和Adams-Cowell法等数值积分方法在轨道预报中已经取得了预期的效果,但是一般不考虑保辛,忽视了系统的固有特性.本文提出适用于轨道预报的乘法保辛摄动方法,将描述卫星运动的Hamilton正则方程分解为二体问题和摄动部分.二体问题采用解析解,摄动部分用区段矩阵近似求解.由于二体问题的状态转移矩阵必然为辛矩阵,故此过程保辛.本文考虑的摄动因素有地球非球形引力、日月引力、太阳光压和潮汐摄动,选取GPS卫星做数值仿真,以GPS卫星精密星历为参照得到轨道预报误差,并与Runge-Kutta法和Adams-Cowell法进行对比.结果表明:对PRN01号GPS卫星和PRN02号GPS卫星进行3 d的轨道预报,本文算法的误差分别为4.56和10.10 m,精度与Runge-Kutta法和Adams-Cowell法一致,而Runge-Kutta法与Adams-Cowell法的计算耗时分别是本文算法的237.7%与71.3%,因此本文算法效率明显高于Runge-Kutta法,但比Adams-Cowell法稍低.
In order to predict artificial satellite orbit accurately, complex dynamic models need to be considered and numerical integration algorithms are used to solve the problem. Desired effect has been obtained for orbit prediction by various integration algorithms such as Runge-Kutta algorithm and Adams-Cowell algorithm. But the symplectic property of the dynamic problem is not taken into consideration, therefore the inherent characteristics of the system are neglected. In this paper, the multiplictive perturbation method based on symplectic propterty, which adapts to orbit prediction, is proposed. Hamiltonian canonical equation, which describes the motion of the satellite, is divided into two sections, including two-body problem and perturbation section. Analytical solution of two-body problem is used. Perturbation section is solved approximately by interval matrices algorithm. Because state transition matrix of the two-body problem is a symplectic matrix, it is a symplectic structure-preserving process. Perturbation factors, including non-spherical Earth, solar-lunar gravitational force, solar radiation pressure and tidal perturbation, are taken into account. GPS satellites are selected to make numerical simulations. The error of orbit prediction is obtained by reference to precise ephemeris of GPS satellites. The results of the proposed perturbation method are compared with that of Runge-Kutta algorithm and Adams-Cowell algorithm. Results show: 3 d prediction errors of proposed algorithm for PRN01 GPS satellite and PRN02 GPS satellite are 4.56 and 10.10 m, respectively. The accuracy of proposed algorithm coincides with that of Runge-Kutta algorithm and Adams-Cowell algorithm. The computational time of Runge-Kutta algorithm is 237.7% of that of proposed algorithm. The computational time of Adams-Cowell algorithm is 71.3% of that of proposed algorithm. So the efficiency of proposed algorithm is obviously higher than that of Runge-Kutta algorithm and is slightly lower than that of Adams-Cowell algorithm.
In order to predict artificial satellite orbit accurately, complex dynamic models need to be considered and numerical integration algorithms are used to solve the problem. Desired effect has been obtained for orbit prediction by various integration algorithms such as Runge-Kutta algorithm and Adams-Cowell algorithm. But the symplectic property of the dynamic problem is not taken into consideration, therefore the inherent characteristics of the system are neglected. In this paper, the multiplictive perturbation method based on symplectic propterty, which adapts to orbit prediction, is proposed. Hamiltonian canonical equation, which describes the motion of the satellite, is divided into two sections, including two-body problem and perturbation section. Analytical solution of two-body problem is used. Perturbation section is solved approximately by interval matrices algorithm. Because state transition matrix of the two-body problem is a symplectic matrix, it is a symplectic structure-preserving process. Perturbation factors, including non-spherical Earth, solar-lunar gravitational force, solar radiation pressure and tidal perturbation, are taken into account. GPS satellites are selected to make numerical simulations. The error of orbit prediction is obtained by reference to precise ephemeris of GPS satellites. The results of the proposed perturbation method are compared with that of Runge-Kutta algorithm and Adams-Cowell algorithm. Results show: 3 d prediction errors of proposed algorithm for PRN01 GPS satellite and PRN02 GPS satellite are 4.56 and 10.10 m, respectively. The accuracy of proposed algorithm coincides with that of Runge-Kutta algorithm and Adams-Cowell algorithm. The computational time of Runge-Kutta algorithm is 237.7% of that of proposed algorithm. The computational time of Adams-Cowell algorithm is 71.3% of that of proposed algorithm. So the efficiency of proposed algorithm is obviously higher than that of Runge-Kutta algorithm and is slightly lower than that of Adams-Cowell algorithm.
引文
1 Abdel-Aziz Y A.An analytical theory for avoidance collision between space debris and operating satellites in LEO.Appl Math Model,2013,37:8283-8291
2 Hsiao H T,Chang T H.Algorithm design for long-term GPS satellite orbit prediction.In:Proceedings of the 2011 Chinese Control and Decision Conference.Mianyang,2011.2761-2766
3管洪杰,姚志成,刘岩.轨道数值积分方法适用性研究.科学技术与工程,2013,36:10883-10886
4 Kosti A A,Anastassi Z A,Simos T E.An optimized explicit Runge-Kutta-Nystr?m method for the numerical solution of orbital and related periodical initial value problems.Comput Phys Commun,2012,183:470-479
5孙雁,高强,钟万勰.保辛-保能的数值积分.应用数学和力学,2014,35:831-837
6 Peng H J,Gao Q,Wu Z G,et al.Symplectic approaches for solving two-point boundary-value problems.J Guid Control Dyn,2012,35:653-658
7徐小明,钟万勰.基于四元数表示的多体动力学系统及其保辛积分算法.应用数学和力学,2014,35:1071-1080
8 Van de Vyver H.A symplectic exponentially fitted modified Runge-Kutta-Nystr?m method for the numerical integration of orbital problems.New Astron,2005,10:261-269
9 Bradley B K,Jones B A,Beylkin G,et al.Bandlimited implicit Runge-Kutta integration for astrodynamics.Celest Mech Dyn Astron,2014,119:143-168
10 Surovik D,Scheeres D.Computational efficiency of symplectic integrators for space debris orbit propagation.In:AIAA/AASAstrodynamics Specialist Conference.Minneapolis,2012
11 Hubaux C,Lema?tre A,Delsate N,et al.Symplectic integration of space debris motion considering several Earth’s shadowing models.Adv Space Res,2012,49:1472-1486
12刘林.航天器轨道理论.北京:国防工业出版社,2000
13 Montenbruck O,Gill E.Satellite Orbits:Models,Methods and Applications.王家松,祝开建,胡小工,译.北京:国防工业出版社,2012
14刘林,胡松杰,曹建峰,等.航天器定轨理论与应用.北京:电子工业出版社,2015
15 Xu G.GPS:Theory,Algorithms and Applications.2nd ed.Berlin:Springer,2007
16王甫红,徐其超,龚学文,等.GAAF在星载GPS实时定轨中的应用研究.武汉大学学报(信息科学版),2014,39:47-51
17童科伟,周建平,何麟书.引力加速度估计函数用于快速准确的引力计算.载人航天,2007,3:59-61
18钟万勰.应用力学的辛数学方法.北京:高等教育出版社,2006
19吴锋.周期结构的时域积分方法及散射分析方法研究.博士学位论文.大连:大连理工大学,2014
20刘林,张强,廖新浩.人卫精密定轨中的算法问题.中国科学A辑,1998,28:848-856
2 Hsiao H T,Chang T H.Algorithm design for long-term GPS satellite orbit prediction.In:Proceedings of the 2011 Chinese Control and Decision Conference.Mianyang,2011.2761-2766
3管洪杰,姚志成,刘岩.轨道数值积分方法适用性研究.科学技术与工程,2013,36:10883-10886
4 Kosti A A,Anastassi Z A,Simos T E.An optimized explicit Runge-Kutta-Nystr?m method for the numerical solution of orbital and related periodical initial value problems.Comput Phys Commun,2012,183:470-479
5孙雁,高强,钟万勰.保辛-保能的数值积分.应用数学和力学,2014,35:831-837
6 Peng H J,Gao Q,Wu Z G,et al.Symplectic approaches for solving two-point boundary-value problems.J Guid Control Dyn,2012,35:653-658
7徐小明,钟万勰.基于四元数表示的多体动力学系统及其保辛积分算法.应用数学和力学,2014,35:1071-1080
8 Van de Vyver H.A symplectic exponentially fitted modified Runge-Kutta-Nystr?m method for the numerical integration of orbital problems.New Astron,2005,10:261-269
9 Bradley B K,Jones B A,Beylkin G,et al.Bandlimited implicit Runge-Kutta integration for astrodynamics.Celest Mech Dyn Astron,2014,119:143-168
10 Surovik D,Scheeres D.Computational efficiency of symplectic integrators for space debris orbit propagation.In:AIAA/AASAstrodynamics Specialist Conference.Minneapolis,2012
11 Hubaux C,Lema?tre A,Delsate N,et al.Symplectic integration of space debris motion considering several Earth’s shadowing models.Adv Space Res,2012,49:1472-1486
12刘林.航天器轨道理论.北京:国防工业出版社,2000
13 Montenbruck O,Gill E.Satellite Orbits:Models,Methods and Applications.王家松,祝开建,胡小工,译.北京:国防工业出版社,2012
14刘林,胡松杰,曹建峰,等.航天器定轨理论与应用.北京:电子工业出版社,2015
15 Xu G.GPS:Theory,Algorithms and Applications.2nd ed.Berlin:Springer,2007
16王甫红,徐其超,龚学文,等.GAAF在星载GPS实时定轨中的应用研究.武汉大学学报(信息科学版),2014,39:47-51
17童科伟,周建平,何麟书.引力加速度估计函数用于快速准确的引力计算.载人航天,2007,3:59-61
18钟万勰.应用力学的辛数学方法.北京:高等教育出版社,2006
19吴锋.周期结构的时域积分方法及散射分析方法研究.博士学位论文.大连:大连理工大学,2014
20刘林,张强,廖新浩.人卫精密定轨中的算法问题.中国科学A辑,1998,28:848-856