Evaluating High-Degree-and-Order Gravitational Harmonics and its Application to the State Predictions of a Lunar Orbiting Satellite

Abstract

In this work, an efficient method with which to evaluate the high-degree-and-order gravitational harmonics of the nonsphericity of a central body is described and applied to state predictions of a lunar orbiter. Unlike the work of Song et al. (2010), which used a conventional computation method to process gravitational harmonic coefficients, the current work adapted a well-known recursion formula that directly uses fully normalized associated Legendre functions to compute the acceleration due to the non-sphericity of the moon. With the formulated algorithms, the states of a lunar orbiting satellite are predicted and its performance is validated in comparisons with solutions obtained from STK/Astrogator. The predicted differences in the orbital states between STK/Astrogator and the current work all remain at a position of less than 1 m with velocity accuracy levels of less than 1 mm/s, even with different orbital inclinations. The effectiveness of the current algorithm, in terms of both the computation time and the degree of accuracy degradation, is also shown in comparisons with results obtained from earlier work. It is expected that the proposed algorithm can be used as a foundation for the development of an operational flight dynamics subsystem for future lunar exploration missions by Korea. It can also be used to analyze missions which require very close operations to the moon.

Flight Dynamics and Navigation for Planetary Missions in Korea: Past Efforts, Recent Status, and Future Preparations

“…Song et al (2010a) reported that the lifetime of a lunar polar orbiter is significantly affected by the different degrees and orders of the lunar gravity model, by gravitational attractions of a third body (specifically the Earth), and by the different orbital inclinations. This software was later updated by Song & Kim (2015) by applying a well-known recursion formula that directly uses fully normalized associated Legendre functions to compute the acceleration because of the non-sphericity of the moon. By implementing this formula, a significant amount of computation time was saved, even with high-degree-andorder harmonic coefficients, and accuracy degradation that can arise during numerical simulations, was prevented (Song & Kim 2015).…”

“…This software was later updated by Song & Kim (2015) by applying a well-known recursion formula that directly uses fully normalized associated Legendre functions to compute the acceleration because of the non-sphericity of the moon. By implementing this formula, a significant amount of computation time was saved, even with high-degree-andorder harmonic coefficients, and accuracy degradation that can arise during numerical simulations, was prevented (Song & Kim 2015). To establish more realistic design concepts, the TLI maneuver was optimized and analyzed by applying a finite thrust model (Song et al 2010b).…”

Evolution of the Selenopotential Model and Its Effects on the Propagation Accuracy of Orbits around the Moon

“…More details regarding the force modeling method, for example, those related to implementation of the recursion algorithm and determination of the inertial acceleration due to the nonsphericity of the central body, expressed in body-fixed Cartesian coordinates, can be found in [54][55][56][57].…”

“…The initial orbital epoch is assumed to be January 01, 2020, 00:00:00 (UTC), corresponding to Korea's first experimental lunar orbiter mission, the launch and subsequent nominal mission of which are set to occur in approximately 2019∼ 2020. To numerically integrate the orbiter's states, the proven performance of KARI's lunar orbit propagator is used [57]. Note that external perturbing forces other than the force due to the lunar gravity field are not additionally considered to focus on the main objectives of the current study.…”

Early Phase Contingency Trajectory Design for the Failure of the First Lunar Orbit Insertion Maneuver: Direct Recovery Options

“…To ensure the successful launch of the KPLO, KARI is now performing extensive trajectory design and analysis activities, including joint efforts between KARI and NASA to validate the trajectory design and navigation performance. Several preliminary design studies have already been conducted for lunar mission trajectory design and analysis (Song et al 2008;Song et al 2009;Song et al 2010;Woo et al 2010;Song et al 2011;Choi et al 2013;Song & Kim 2015) as well as orbit determination analysis Lee et al 2017). However, most of the previous work was mainly focused on the optimal trajectory generation from the nominal trajectory planning point of view.…”