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.

Evolutionary approach for spatial architecture layout design enhanced by an agent-based topology finding system

This paper presents a method for the automatic generation of a spatial architectural layout from a user-specified architectural program. The proposed approach binds a multi-agent topology finding system and an evolutionary optimization process. The former generates topology satisfied layouts for further optimization, while the latter focuses on refining the layouts to achieve predefined architectural criteria. The topology finding process narrows the search space and increases the performance in subsequent optimization. Results imply that the spatial layout modeling and the muLti-floor topology are handled.

Constructing Invariant Tori for the Spatial Hill Lunar Problem

In this paper, the spatial Hill lunar problem is investigated, and the existence of invariant tori of hyperbolic type in a neighborhood of its equilibrium is shown. Moreover,the author checks the non-degenerate condition analytically and obtains two-dimensional elliptic invariant tori on its central manifold as well.

Analytic intermediate dimensional elliptic tori for the planetary many-body problem

In our context,the planetary many-body problem consists of studying the motion of(n+1)-bodies under the mutual attraction of gravitation,where n planets move around a massive central body,the Sun.We establish the existence of real analytic lower dimensional elliptic invariant tori with intermediate dimension N lies between n and 3n-1 for the spatial planetary many-body problem.Based on a degenerate KolmogorovArnold-Moser(abbr.KAM)theorem proved by Bambusi et al.(2011),Berti and Biasco(2011),we manage to handle the difficulties caused by the degeneracy of this real analytic system.