Kustaanheimo Stiefel Variables Research Articles

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.

Quaternion Equations of Disturbed Motion of an Artificial Earth Satellite

The quaternion equations of disturbed motion of an artificial Earth satellite (AES) in Earth’s gravitational field (with regard to its zonal, tesseral, and sectorial harmonics) are obtained in four-dimensional Kustaanheimo–Stiefel variables and in modified four-dimensional variables. In the latter variables the equations of satellite motion have a simpler and more symmetric structure, as compared to the equations of motion in the Kustaanheimo–Stiefel variables. The obtained equations are linear for undisturbed Keplerian motion and, in general, have the form of equations of motion of a disturbed four-dimensional oscillator. These equations, in contrast to classical equations, are regular (they do not contain specific points of singularity) for satellite motion in the central gravitational field of Earth under an effect of disturbing forces, the description of which does not contain negative, higher-than-first powers of the satellite distance to Earth’s center. In these equations, the basic variables are Kustaanheimo–Stiefel variables, or modified four-dimensional variables proposed by the author, as well as satellite motion energy and time. The new independent variable is associated with time by the differential relation containing the satellite distance to Earth’s center of mass. The equations are convenient for applying nonlinear mechanics methods and for high-precision numerical calculations. In the case of artificial satellite motion in Earth’s gravitational field, the description of which does not take into account the tesseral and sectorial harmonics, but takes into account its zonal harmonics, the first integrals of the equations of satellite motion are found, and the replacements of variables and transformations of these equations are proposed, which allowed one to get, for studying satellite motion, closed systems of differential equations of lower dimension, in particular, the system of equations of fourth and third orders.

Inertial Navigation in Space Using the Regular Quaternion Equations of Astrodynamics

Quaternion equations are proposed for the ideal operation of spatial inertial navigation systems with an azimuthally stabilized platform and a gyrostabilized platform that keeps its orientation invariant in inertial space, quaternion equations for the ideal operation of strapdown inertial navigation systems in the regular four-dimensional Kustaanheimo-Stiefel variables with consideration of zonal, tesseral, and sectorial harmonics of the Earth's gravitational field. The equations are dynamically similar to the regular equations of perturbed spatial two-body problem in the Kustaanheimo-Stiefel variables, which enables to use the results obtained in regular celestial mechanics and astrodynamics theory in inertial astronavigation. The development of operational algorithms for these navigation systems using these equations is considered.

Geometric chaos indicators and computations of the spherical hypertube manifolds of the spatial circular restricted three-body problem

The circular restricted three-body problem has five relative equilibria L1,L2, ...,L5. The invariant stable–unstable manifolds of the center manifolds originating at the partially hyperbolic equilibria L1,L2 have been identified as the separatrices for the motions which transit between the regions of the phase-space which are internal or external with respect to the two massive bodies. While the stable and unstable manifolds of the planar problem have been extensively studied both theoretically and numerically, the spatial case has not been as deeply investigated. This paper is devoted to the global computation of these manifolds in the spatial case with a suitable finite time chaos indicator. The definition of the chaos indicator is not trivial, since the mandatory use of the regularizing Kustaanheimo–Stiefel variables may introduce discontinuities in the finite time chaos indicators. From the study of such discontinuities, we define geometric chaos indicators which are globally defined and smooth, and whose ridges sharply approximate the stable and unstable manifolds of the center manifolds of L1,L2. We illustrate the method for the Sun–Jupiter mass ratio, and represent the topology of the asymptotic manifolds using sections and three-dimensional representations.

Quaternion Regularization of the Equations of the Perturbed Spatial Restricted Three-Body Problem: I

We develop a quaternion method for regularizing the differential equations of the perturbed spatial restricted three-body problem by using the Kustaanheimo–Stiefel variables, which is methodologically closely related to the quaternion method for regularizing the differential equations of perturbed spatial two-body problem, which was proposed by the author of the present paper. A survey of papers related to the regularization of the differential equations of the two- and threebody problems is given. The original Newtonian equations of perturbed spatial restricted three-body problem are considered, and the problem of their regularization is posed; the energy relations and the differential equations describing the variations in the energies of the system in the perturbed spatial restricted three-body problem are given, as well as the first integrals of the differential equations of the unperturbed spatial restricted circular three-body problem (Jacobi integrals); the equations of perturbed spatial restricted three-body problem written in terms of rotating coordinate systems whose angular motion is described by the rotation quaternions (Euler (Rodrigues–Hamilton) parameters) are considered; and the differential equations for angular momenta in the restricted three-body problem are given. Local regular quaternion differential equations of perturbed spatial restricted three-body problem in the Kustaanheimo–Stiefel variables, i.e., equations regular in a neighborhood of the first and second body of finite mass, are obtained. The equations are systems of nonlinear nonstationary eleventhorder differential equations. These equations employ, as additional dependent variables, the energy characteristics of motion of the body under study (a body of a negligibly small mass) and the time whose derivative with respect to a new independent variable is equal to the distance from the body of negligibly small mass to the first or second body of finite mass. The equations obtained in the paper permit developing regular methods for determining solutions, in analytical or numerical form, of problems difficult for classicalmethods, such as the motion of a body of negligibly small mass in a neighborhood of the other two bodies of finite masses.

Energy preserving low-thrust guidance for orbit transfers in KS variables

A Lyapunov-based approach is presented to design finite-thrust closed-loop guidance schemes in a two-body model, to perform any general orbit transfer. The design is performed in the Kustaanheimo–Stiefel (KS) model, a regularized two-body framework. The KS model aids in the design of the guidance schemes, but the new guidance solutions can be explicitly characterized in the regular two-body model. The control scheme covers two main phases: first, a matching of the semi-major axis of the target orbit; and second, a matching of the remaining desired orbital elements by holding energy constant. No constraint is imposed on the magnitude of the thrust; however, an emphasis is placed on low-thrust examples.

New Solutions for the Perturbed Lambert Problem Using Regularization and Picard Iteration

A new approach for solving two-point boundary value problems and initial value problems using the Kustaanheimo–Stiefel transformation and Modified Chebyshev–Picard iteration is presented. The first contribution is the development of an analytical solution to the elliptic Keplerian Lambert problem based on Kustaanheimo–Stiefel regularization. This transforms the nonlinear three-dimensional orbit equations of motion into four linear oscillators. The second contribution solves the elliptic Keplerian two-point boundary value problem and initial value problem using the Kustaanheimo–Stiefel transformation and Picard iteration. The Picard sequence of trajectories represents a contraction mapping that converges to a unique solution over a finite domain. Solving the Keplerian two-point boundary value problem in Kustaanheimo–Stiefel variables increases the Picard domain of convergence from about one-third of an orbit (Cartesian variables) to over 95% of an orbit (Kustaanheimo–Stiefel variables). These increases in the domain of Picard iteration convergence are independent of eccentricity. The third contribution solves the general spherical harmonic gravity perturbed elliptic two-point boundary value problem using the Kustaanheimo–Stiefel transformation and Picard iteration, and it does not require a Newton-like shooting method for fractional orbit transfers. For multiple revolution transfers, however, a shooting method can make use of the Modified Chebyshev–Picard iteration/Kustaanheimo–Stiefel/initial value problem and the Method of Particular Solutions to obtain solutions given a Keplerian Lambert solution as the starting iterative. The Kustaanheimo–Stiefel perturbed solution is illustrated using a (40,40) degree and order spherical harmonic gravity model. A general three-dimensional recipe is introduced for solving the perturbed Lambert Problem via Modified Chebyshev–Picard iteration without a Newton-like shooting method for the fractional orbit case. The increase in the domain of convergence of the Kustaanheimo–Stiefel transformed, perturbed Lambert problem via Modified Chebyshev–Picard iteration versus the Cartesian Modified Chebyshev–Picard iteration Lambert solution is analogous to the results for the Keplerian case. The three-dimensional two-impulse perturbed Lambert problem is efficiently convergent up to about 85% of the Keplerian orbit period with a (40,40) spherical harmonic gravity model. The efficiency of the current two-point boundary value problem solver is compared with MATLAB’s fsolve, where Runge–Kutta–Nystrom 12(10) and Gauss–Jackson are the integrators.

KS variables in rotating reference frame. Application to cometary dynamics

The problem of Keplerian motion in the uniformly rotating reference frame has been solved in terms of the Kustaanheimo-Stiefel (KS) variables using canonical formalism. No recourse to the Cartesian variables or orbital elements has been required. The form of solution is well suited for the application as a part of symplectic integrator. The results show that the motion is actually the composition of four independent harmonic oscillations and of the rotation in two specific coordinate planes and their conjugate momenta planes. As an example of application, we use the KS symplectic integrator to study the motion of comet C/1997 J2 (Meunier-Dupuoy) under the action of Galactic tides. The comet is found to follow an orbit in commensurability with the Sun motion around the Galactic centre, but the perturbations are not qualified as a resonance.

Construction of optimum controls and trajectories of motion of the center of masses of a spacecraft equipped with the solar sail and low-thrust engine, using quaternions and Kustaanheimo-Stiefel variables

The problem of optimum rendezvous of a controllable spacecraft (SC) with an uncontrollable spacecraft, moving over a Keplerian elliptic orbit in the gravitational field of the Sun, is considered. Control of the SC is performed using a solar sail and low-thrust engine. For solving the problem, the regular quaternion equations of the two-body problem with the Kustaanheimo-Stiefel variables and the Pontryagin maximum principle are used. The combined integral quality functional, which characterizes energy consumption for controllable SC transition from an initial to final state and the time spent for this transition, is used as a minimized functional. The differential boundary-value optimization problems are formulated, and their first integrals are found. Examples of numerical solution of problems are presented. The paper develops the application [1–6] of quaternion regular equations with the Kustaanheimo-Stiefel variables in the space flight mechanics.

Collective branch regularization of simultaneous binary collisions in the 3D N-body problem

In this work we study simultaneous binary collision (SBC) singularities of M⩽N∕2 binaries in the three dimensional classical gravitational N-body problem. We show the following: (1) In the generalized Kustaanheimo–Stiefel variables, the totality of SBC orbits, the totality of simultaneous binary collisions (SBE) orbits, and the collision singularity itself together form a real analytic submanifold which we call the collision-ejection manifold. (2) We use the collision-ejection manifold to show geometrically, without writing down any power series, that SBC solutions can be collectively analytically continued. That is, all SBC orbits, not just a single orbit, can be written as a convergent power series in s=t1∕3 with coefficients that depend real analytically on initial conditions that lie in a real analytic submanifold. (3) There are two important ingredients in our work. (i) We use the intrinsic energies and properly rescaled intrinsic angular momenta of the binaries as variables in order to reduce the order of the singularity and to parametrize (distinguish between different) collision orbits that constitute the stable manifolds of the rest points that appear on the collision manifold in the McGehee coordinates. (ii) We use what we call the Kustaanheimo–Stiefel-projective transformation near a SBC singularity to resolve the singularity and isolate collision and ejection orbits from nearby near-collision and near-ejection orbits. We will see that quaternionic multiplication and quaternionic projective spaces are not suitable.

Traitement algébrique du système de MIC–Kepler

The MIC–Kepler system is studied via the Milshtein and Strakhovenko variant of the so(2,1) Lie algebra. The Green function is constructed in parabolic coordinates, with the help of the Kustaanheimo–Stiefel variables and the generators of the SO(2,1) group. The energy spectrum and the normalized wave functions of the bound states are obtained.

Algebraic treatment of the Kaluza–Klein monopole system

In analogy to the Kepler problem, the Green’s function for a particle in the background of the Kaluza–Klein monopole is constructed through the algebraic approach, with the help of the Kustaanheimo–Stiefel variables and the generators of the SO(2,1) group in the exponential representation of Schwinger. The bound and continuum states are also obtained.