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Quaternion methods and models of regular celestial mechanics and astrodynamics

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Applied Mathematics and Mechanics(English Edition) 2022年第1期43卷 21-80页

作者： Y.N.CHELNOKOV Institute of Precision Mechanics and Control Russian Academy of SciencesSaratov410031Russia

This paper is a review,which focuses on our work,while including an analysis of many works of other researchers in the field of quaternionic regularization.The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel(KS)variables and Euler(Rodrigues-Hamilton)parameters are analyzed.These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle.This paper also covers some applications of these models.Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics.However,the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author.More general(compared with the KS equations)quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented.These equations are derived with the assumption that the KS bilinear relation was not satisfied.The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented,together with regular equations in the KS variables and Euler parameters,derived by the aforementioned theory.We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters,developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.

关键词： celestial mechanics astrodynamics perturbed spatial two-body problem central motion regularization Levi-Civita and Kustaanheimo-Stiefel(KS)variables Euler parameter quaternion

A review of space-object collision probability computation methods

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astrodynamics 2022年第2期6卷 95-120页

作者： Jia-Sheng Li Zhen Yang Ya-Zhong Luo College of Aerospace Science and Engineering National University of Defense TechnologyChangsha 410073China Hunan Key Laboratory of Intelligent Planning and Simulation for Aerospace Missions Changsha 410073China

The collision probability computation of space objects plays an important role in space situational awareness,particularly for conjunction assessment and collision avoidance.Early works mainly relied on Monte Carlo simulations to predict collision probabilities.Although such simulations are accurate when a large number of samples are used,these methods are perceived as computationally intensive,which limits their application in practice.To overcome this limitation,many approximation methods have been developed over the past three decades.This paper presents a comprehensive review of existing space-object collision probability computation methods.The advantages and limitations of different methods are analyzed and a systematic comparison is presented.Advice regarding how to select a suitable method for different short-term encounter scenarios is then provided.Additionally,potential future research avenues are discussed.

关键词： collision probability space situational awareness collision avoidance astrodynamics

Inverse Problem of astrodynamics

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World Journal of Mechanics 2015年第12期5卷 249-256页

作者： Yuri Menshikov Department of Mechanics and Mathematics Dnepropetrovsk University Dnepropetrovsk Ukraine

We consider the problem of determining the center of mass of an unknown gravitational body, using the disturbances in the motion of observed celestial bodies. In this paper an universal approach to obtain the approximate and stable estimate of problem solution is suggested. This approach can be used in other fields of Science. For example, it can be applied for investigation of interactions between fields of forces and elementary particles using known trajectories of elementary particles motions.

关键词： astrodynamics Le Verrier Problem Regularization Main Hypothesis

Evolutionary Algorithms in astrodynamics

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International Journal of Astronomy and Astrophysics 2016年第4期6卷 435-439页

作者： Marcin Misiak UAM Poznan Poland

This paper presents the method of solving the equations of motions by evolutionary algorithms. Starting from random trajectory, the solution is obtained by accepting the mutation if it leads to a better approximations... 详细信息

关键词： Genetic Algorithms Newton’s Equations of Motion Principle of Least Action astrodynamics Celestial Mechanics Orbital Motion Optimal Control Two-Point Boundary Value Problem

Artificial equilibrium points and bi-impulsive maneuvers to observe 243 Ida

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Chinese Journal of Aeronautics 2021年第1期34卷 410-423页

作者： Allan Kardec de ALMEIDA Jr Geraldo Magela Couto OLIVEIRA Antonio Fernando Bertachini de Almeida PRADO National Institute for Space Research(INPE) Sao Jose dos Campos 12227-010Brazil Federal Center for Technological Education of Minas Gerais(CEFET-MG) Belo Horizonte 30421-169Brazil

In the restricted three-body problem,the traditional Lagrange points L1 and L2 are the only equilibrium points near the asteroid 243 Ida.The thrust generated by a solar sail over a spacecraft enables the existence of new artificial equilibrium points,which depend on the position of the spacecraft with respect to the asteroid and the attitude of the solar sail.Such equilibrium points generate new spots to observe the body from above or below the plane of motion.Such points are very good observational locations due to their stationary condition.This work provides a preliminary analysis to observe Ida through the use of artificial equilibrium points as spots combined with transfer maneuvers between them.Such combination can be used to observe the asteroid from more different points of view in comparison to fixed ones.The analyses are made for a spacecraft equipped with a solar sail and capable of performing bi-impulsive maneuvers.The solar radiation pressure is used both to maintain the equilibrium condition and to reduce the costs of the transfers and/or to create transfers with longer duration.This is a new aspect of the present research,because it combines the continuous thrust with initial and final small impulses,which are feasible for most of the spacecraft,because the magnitudes of the impulses are very low.These combined maneuvers may reduce the transfer times of the maneuvers in most of the cases,compared with the maneuvers based only on continuous thrust.Several options involved in these transfers are shown,like to minimize the fuel spent(Dv)as a function of the transfer time or to extend the duration of the travel between the points.Extended transfer times can be useful when observations are required during the transfers.

关键词： Artificial equilibrium points astrodynamics Orbital transfers Solar radiation pressure Solar sail

SmallSat swarm gravimetry:Revealing the interior structure of asteroids and comets

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astrodynamics 2021年第3期5卷 217-236页

作者： William Gordon Ledbetter Rohan Sood James Keane Jeffrey Stuart astrodynamics and Space Research Laboratory The University of AlabamaTuscaloosaAL 35487USA Jet Propulsion Laboratory California Institute of TechnologyPasadenaCA 91109USA

A growing interest in small body exploration has motivated research into the rapid char-acterization of near-Earth objects to meet economic or scientific objectives.Specifically,knowledge of the internal density structure can aid with target selection and enables an understanding of prehistoric planetary formation to be developed.To this end,multi-layer extensions to the polyhedral gravity model are suggested,and an inversion technique is implemented to present their effectiveness.On-orbit gravity gradiometry is simulated and employed in stochastic and deterministic algorithms,with results that imply robustness in both cases.

关键词： gravimetry asteroid exploration astrodynamics stochastic optimization

Enhancements to Modified Chebyshev-Picard Iteration Efficiency for Perturbed Orbit Propagation

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Computer Modeling in Engineering & Sciences 2016年第1期111卷 29-64页

作者： B.Macomber A.B.Probe R.Woollands J.Read J.L.Junkins Texas A&M University College StationTXUSA.

Modified Chebyshev Picard Iteration is an iterative numerical method for solving linear or non-linear ordinary differential equations.In a serial computational environment the method has been shown to compete with,or outperform,current state of practice numerical integrators.This paper presents several improvements to the basic method,designed to further increase the computational efficiency of solving the equations of perturbed orbit propagation.

关键词： astrodynamics Orbit Propagation Cheybshev Polynomials Picard Iteration.

Design and flight results of GNC systems in Hayabusa2 descent operations

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astrodynamics 2020年第2期4卷 105-117页

作者： Go Ono Fuyuto Terui Naoko Ogawa Yuya Mimasu Kent Yoshikawa Yuto Takei Takanao Saiki Yuichi Tsuda Japan Aerospace Exploration Agency Sagamihara252-5210Japan

Hayabusa2 is a Japanese sample return mission from the near-Earth asteroid Ryugu.The Hayabusa2 spacecraft was launched on December 3,2014,and reached the asteroid on June 27,2018.It remained there until November 13,2019 for in situ observation and soil sample collection and will return to the Earth in November or December 2020.During its stay at the asteroid,Hayabusa2 performed descent operations 16 times.This paper presents an overview of a guidance,navigation,and control method used in such descent operations.The method consists of on-board and on-ground guidance systems to control the spacecraft and an image-based navigation technique that uses a shape model and ground control points of the asteroid.Flight results in the first touchdown operation are shown as an example,which demonstrate that the method showed a good performance overall and contributed to the success of the mission.

关键词： small body exploration astrodynamics guidance navigation control

Parametric variational solution of linear-quadratic optimal control problems with control inequality constraints

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Applied Mathematics and Mechanics(English Edition) 2014年第9期35卷 1079-1098页

作者：彭海军高强张洪武吴志刚钟万勰 State Key Laboratory of Structural Analysis for Industrial Equipment Department of Engineering MechanicsDalian University of Technology State Key Laboratory of Structural Analysis for Industrial Equipment School of Aeronautics and AstronauticsDalian University of Technology

A parametric variational principle and the corresponding numerical algo- rithm are proposed to solve a linear-quadratic （LQ） optimal control problem with control inequality constraints. Based on the parametric variational principle, this control prob- lem is transformed into a set of Hamiltonian canonical equations coupled with the linear complementarity equations, which are solved by a linear complementarity solver in the discrete-time domain. The costate variable information is also evaluated by the proposed method. The parametric variational algorithm proposed in this paper is suitable for both time-invariant and time-varying systems. Two numerical examples are used to test the validity of the proposed method. The proposed algorithm is used to astrodynamics to solve a practical optimal control problem for rendezvousing spacecrafts with a finite low thrust. The numerical simulations show that the parametric variational algorithm is ef- fective for LQ optimal control problems with control inequality constraints.

关键词： parametric variational principle optimal control inequality constraint,linear complementarity astrodynamics linear-quadratic （LQ）