Quaternion methods and models of regular celestial mechanics and astrodynamics

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.

Inverse Problem of Astrodynamics

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.

Evolutionary Algorithms in Astrodynamics

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 of Newton’s second law. The general method is illustrated by finding trajectory to the Moon.

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

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.

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

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.

Solution of Liouville’s Equation for Uncertainty Characterization of the Main Problem in Satellite Theory

This paper presents a closed form solution to Liouville’s equation governing the evolution of the probability density function associated with the motion of a body in a central force field and subject to J2.It is shown that the application of transformation of variables formula for mapping uncertainties is equivalent to the method of characteristics for computing the time evolution of the probability density function that forms the solution of the Liouville’s partial differential equation.The insights derived from the nature of the solution to Liouville’s equation are used to reduce the dimensionality of uncertainties in orbital element space.It is demonstrated that the uncertainty propagation is fastest in the semi-major axis and the mean anomaly phase sub-space.The results obtained for uncertainty propagation for the two body problem are applied to investigate the uncertainty propagation in the presence of the J2 perturbation using a combination of osculating and mean element perturbation theory.Analytical orbital uncertainty propagation calculations are validated using Monte-Carlo results for several representative orbits.

Mass Density Distributions in Spiral Galaxies

The purpose of this paper is to show, on the basis of Newtonian mechanics (in Euclidean space), that the core disks of spiral galaxies (the central disks in galactic cores that are perpendicular to the axes of rotation) rotate in the same fashion as a phonograph turntable, if the mass densities in the cores of such galaxies remain uniform. On the basis of the hypothesis of uniform mass density in the core, it is then shown that the density of mass in the shell (the entire domain outside of the core) must remain inversely proportional to the square of radial distance from the axis of rotation and that the angular velocity in the shell annulus (annulus in the shell that contains the spiral forms) is inversely proportional to radial distance, or that the circumferential velocity on the shell disk is independent of radial distance from the core axis. The equation of motion for the shell disk is then obtained and it is concluded that the spiral shaped lanes are not trajectories. But it is shown that any bar-shaped feature crossing the shell annulus and core disk, collinear with the core centre, will become distorted, due to the above angular velocity distribution in the shell disk, assuming the form of two, symmetrically disposed, Archimedean spirals, while the portion of the bar inside the core remains undistorted and merely rotates.

The Solution of Optimal Two-Impulse Transfer between Elliptical Orbits with Plane Change

The optimizing total velocity increment Δv needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velocities concept due to the changing in the energy. The transferring has been made between two elliptic orbits having a common centre of attraction with changing in their planes in standard Hohmann transfer with the terminal orbit which is elliptic orbit and not circular. We develop a treatment based on the elements of elliptic orbits a1,e1, a2,e2, and?aT,eT of the initial orbit, final orbit and transferred orbit respectively. The first impulse Δv1 at the perigee induces a rotation of the orbital plane by ?which will be minimized. The second impulse Δv2 at apogee is induced an angle ?to product the final elliptic orbit. The total plane change required . We calculate the total impulse Δv and minimize by optimizing angle of plane’s variation . We obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits ?and . The solution obtained numerically, using programming code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. We find that there are constrains on the transfer angles and α. For αit must be between 40°and 160°, and there is no solution if αis less than 40°and bigger than 160°and ?takes the values less than 40°. The minimum total velocity increments obtained at the value of ?less than 25°and& alpha;equal to 160°. This is an interesting result in orbital transfer problem in which the change of orbital plane is necessary for the transferring.

Cosmic Engineering: Moving Asteroids

This paper presents methods of orbit transfer for small planetoids from the main belt to the future colonies on Mars using current technologies. The results show that by using nuclear weapon or even kinetic energy weapon (for retrograde bodies) asteroids with masses up to about 100 tons can be moved. Both options assume that asteroid will survive explosion.

Solar Radiation Pressure and Gravitational Waves Effects on Sun-Synchronous Orbits

For certain values of semi-major axis and eccentricity, orbit plane precession caused by Earth oblate is synchronous with the mean orbital motion of the apparent Sun (a sun-synchronism). Many forces cause slow changes in the inclination and ascending node of sun-synchronous orbits. In this work, we investigate the analytical perturbations due to the direct solar radiation pressure and gravitational waves effects. A full analytical solution is obtained using technique of canonical Lie-transformation up to the order three in (the oblateness of the Earth). The solar radiation pressure and gravitational waves perturbations cause second order effects on all the elements of the elliptic orbit (the eccentricity, inclination, ascending node, argument of perigee, and semi-major axis) consequently these perturbations will cause disturbance in the sun-synchronism. Also we found that the perturbation or the behavior of gravitational waves almost the same as the perturbation or the behavior of solar radiation pressure and their coupling will incorporate the sun-synchronism through the secular rate of the ascending node precession.

Proposal of New Criteria for Celestial Mechanics

Based on a new interpretation on the behavior of rigid bodies exposed to simultaneous non-coaxial rotations, we have developed a hypothesis: the Theory of Dynamics Interactions, which can be applied to understand celestial mechanics. We have analyzed the velocity and acceleration fields generated in a rigid body with intrinsic angular momentum, when exposed to successive torques, to assess new criteria for this speeds coupling. In this context, reactions and inertial fields take place, which cannot be justified by means of classical mechanics. We believe that the results obtained after the analysis of dynamics fields systems accelerated by rotation will allow us to conceive a new perspective in celestial dynamics, astrometry, stellar dynamics and galactic astronomy, unknown up to date. After carrying out ample research, we have come to the conclusion that there still exists an unstructured scientific area under the present general assumptions and, more specifically, in the area of dynamic systems submitted to rotational accelerations. The aim of this paper is to present information of the surprising results obtained, and to attract the interest towards the investigation of this new area of knowledge in rotational non-inertial dynamics, and its multiple and remarkable scientific applications.

How are multiple satellites seen from the ground? Relative apparent motion and formation stabilization

This paper answers how multiple satellites are seen from the ground. This question is inspired by space-advertising, a public exhibition in the night sky using a dot matrix of satellites that are bright enough to be seen by the naked eye. Thus, it is important for space advertisement that the specific dot matrix is seen. Moreover, the stability of the dot matrix during a visible span is very valuable. To stabilize the dot matrix, this study formulates an apparent position of a dot from a representative dot seen from the ground. The formulation, linear functions of a set of relative orbital elements, reveals the appearance of the dot matrix. The proposed relative variable in the formulation drives the instability of the dot matrix, thereby revealing an initial stable configuration of deputies from a chief. The arbitrary dot matrix designed using the configuration is stable even at low elevations without orbital control during the visible span.

A review of space-object collision probability computation methods

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.

Modeling and analysis of Hayabusa2 touchdown

The Hayabusa2 asteroid explorer mission focuses principally on the touchdown and sampling on near-Earth asteroid 162173 Ryugu.Hayabusa2 successfully landed on its surface and ejected a projectile for sample collection on February 22,2019.Hayabusa2 later landed near a crater formed by an impactor and executed the sampling sequence again on July 11,2019.For a successful mission,a thorough understanding and evaluation of spacecraft dynamics during touchdown were crucial.The most challenging aspect of this study was the modeling of such spacecraft phenomena as the dynamics of landing on a surface with unknown properties.In particular,a Monte Carlo analysis was used to determine the parameters of the operational design for the final descent and touchdown sequence.This paper discusses the dynamical modeling of the simulation during the touchdown of Hayabusa2.

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

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.

Guidance, navigation, and control of Hayabusa2 touchdown operations

Hayabusa2 is a Japanese sample return mission from the asteroid Ryugu.The Hayabusa2 spacecraft was launched on 3 December 2014 and arrived at Ryugu on 27 June 2018.It stayed there until December 2019 for in situ observation and soil sample collection,and will return to the Earth in November or December 2020.During the stay,the spacecraft performed the first touchdown operation on 22 February 2019 and the second touchdown on 11 July 2019,which were both completed sucssfully.Because the surface of Ryugu is rough and covered with boulders,it was not easy to find target areas for touchdown.There were several technical challenges to overcome,including demanding guidance,navigation,and control accuracy,to realize the touchdown operation.In this paper,strategies and technical details of the guidance,navigation,and control systems are presented.The flight results prove that the performance of the systems was satisfactory and largely contributed to the success of the operation.

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

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.

Design and flight results of GNC systems in Hayabusa2 descent operations

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.