Analytical guidance in the neighborhood of optimal multi-impulse trajectories

A first-order minimum propellant guidance law is developed for multi-impulse trajectories in an inverse-square gravitational field. A second-order variational analysis is used to formulate the guidance problem as an accessory minimum problem, i.e. minimize a quadratic form (second-variation in propellant consumption) subject to linear constraints (variational equations of motion and deterministic boundary conditions). Solution of the accessory minimum problem provides the optimal guidance law in feedback form. It is emphasized that this analysis takes into account the nominal impulse programme when calculating the optimal guidance corrections. It is shown that for multi-impulse transfers it is in general, non-optimal to add impulses. All corrections to the trajectory should be made by a combination of small changes in timing, magnitude and direction of the nominal impulses.     

A multi-objective approach to the design of low thrust space trajectories using optimal control

In this article, we introduce a novel three-step approach for solving optimal control problems in space mission design. We demonstrate its potential by the example task of sending a group of spacecraft to a specific Earth L ₂ halo orbit. In each of the three steps we make use of recently developed optimization methods and the result of one step serves as input data for the subsequent one. Firstly, we perform a global and multi-objective optimization on a restricted class of control functions. The solutions of this problem are (Pareto-)optimal with respect to ΔV and flight time. Based on the solution set, a compromise trajectory can be chosen suited to the mission goals. In the second step, this selected trajectory serves as initial guess for a direct local optimization. We construct a trajectory using a more flexible control law and, hence, the obtained solutions are improved with respect to control effort. Finally, we consider the improved result as a reference trajectory for a formation flight task and compute trajectories for several spacecraft such that these arrive at the halo orbit in a prescribed relative configuration. The strong points of our three-step approach are that the challenging design of good initial guesses is handled numerically by the global optimization tool and afterwards, the last two steps only have to be performed for one reference trajectory.     

On target for Venus – set oriented computation of energy efficient low thrust trajectories

Recently new techniques for the design of energy efficient trajectories for space missions have been proposed that are based on the circular restricted three body problem as the underlying mathematical model. These techniques exploit the structure and geometry of certain invariant sets and associated invariant manifolds in phase space to systematically construct energy efficient flight paths. In this paper, we extend this model in order to account for a continuously applied control force on the spacecraft as realized by certain low thrust propulsion systems. We show how the techniques for the trajectory design can be suitably augmented and compute approximations to trajectories for a mission to Venus.     

A comparative study of Newtonian,Kustaanheimo/Stiefel,and Sperling/Burdet optimal trajectories

An optimal trajectory problem is formulated in each of three sets of equations, and the resulting solutions are numerically compared. The three formulations are the classical Newtonian (N), the Kustaanheimo/Stiefel (K/S), and the Sperling/Burdet (S/B). The last two solutions are first regularized by the classical Sundman technique and the K/S solution is transformed before the optimization problem is posed. A novel technique is developed for generating initial control vectors for each solution. Numerically generated derivatives (central differences) are used by a type of gradient, Newton-Raphson iterator to converge the two-point boundary value problems. The results indicate that, although the K/S and S/B formulations are more difficult to express mathematically than the Newtonian formulation, the transformed solutions are significantly more numerically stable than the Newtonian solution when the perturbing acceleration is less than a minimum value (T/W _o=0.05 for the particular example problem treated).     

Regularization and the artificial Earth satellite problem

Some of the properties of singularities of a system of differential equations, which includes several formulations of the artificial Earth satellite problem, are derived. Using them, it is shown that this problem cannot be regularized by using the current commonly used ideas and definitions of regularization.     

Regularization of the equatorial magnetic-binary problem

The differential equations which describe the equatorial motion of a particle in a magnetic-binary system are regularized by a transformation of the dependent and independent variables. The new equations of motion, which are simpler than the original ones, with only linear velocity and acceleration terms occurring, show the invariant form of the law of motion. Finally, the distinction between the zero-velocity curves and the isotachs in the transformed space is also discussed.     

Regularization in the Ideal Resonance Problem

The Ideal Resonance Problem, defined by the HamiltonianF=B(y)+2A (y) sin² x, 1, has been solved in Garfinkelet al. (1971). There the solution has beenregularized by means of a special function _j , introduced into the new HamiltonianF, under the tacit assumption thatA anB¨' are of order unity.This assumption, violated in some applications of the theory, is replaced here by the weaker assumption ofnormality, which admits zeros ofA andB inshallow resonance. It is shown here that these zeros generate singularities, which can be suppressed if _j is suitably redefined.With the modified _j , and with the assumption of normality, the solution is regularized for all values ofB, B¨', andA. As in the previous paper, the solution isglobal, including asymptotically the classical limit withB as a divisor of O(1).A regularized first-order aloorithm is constructed here as an illustration and a check.Presented at the XXII International Congress of I.A.F., Brussels, Belgium, Sept. 20, 1971.     

Regularization and linearization of the equations of motion in central force-fields

Within the framework of linear and regular celestial mechanics, a wide class of central force field problems are considered. We take as potential function a polynomial whose variable is the reciprocal of the distance from the origin, and, as regularizing function the square root of a certain reciprocal polynomial, related to the potential function. The relations between the coefficients of both functions, in order to arrive to linear equations, are given. In particular, the case of a quintic polynomial is analyzed, and an application is made to artificial earth satellites considering harmonicsJ ₂,J ₃,J ₄.     

Global Regularization of the Restricted Problem of Three Bodies

The global regularizing transformations of the planar, circular restricted problem of three bodies are studied. It is shown that all these transformations can be written in the same general form which is the solution of a first order ordinary differential equation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Ordered and chaotic Bohmian trajectories

We discuss the issue of ordered and chaotic trajectories in the Bohmian approach of Quantum Mechanics from points of view relevant to the methods of Celestial Mechanics. The Bohmian approach gives the same results as the orthodox (Copenhagen) approach, but it considers also underlying trajectories guided by the wave. The Bohmian trajectories are rather different from the corresponding classical trajectories. We give examples of a classical chaotic system that is ordered quantum-mechanically and of a classically ordered system that is mostly chaotic quantum mechanically. Then we consider quantum periodic orbits and ordered orbits, that can be represented by formal series of the “third integral” type, and we study their asymptotic properties leading to estimates of exponential stability. Such orbits do not approach the “nodal points” where the wavefunction ψ vanishes. On the other hand, when an orbit comes close to a nodal point, chaos is generated in the neighborhood of a hyperbolic point (called X-point). The generation of chaos is maximum when the X-point is close to the nodal point. Finally we remark that high order periodic orbits may behave as “effectively ordered” or “effectively chaotic” for long times before reaching the period.     

A Note on the Regularization of the Restricted Three-body Problem

The requirement that near a singular point of the equations of motion the power series expansions of the old variables in terms of the new ones start with second order terms leads to the transformation z = sin²1/2w related to that of THIELE -BURRAU . Using this new transformation, a derivation of the regularized equations of motion is given. The original as well as the regularized equations of motion are of interest, for example, for calculating the initial values of the orbital elements for SCHWARZSCHILD's periodic solutions (LEIMANIS and OLUND 1972).     

Regularization of the two body problem with variable mass

Using Levi Civita's regularization, we put the two body problem with variable mass (x=?Mxr ^?3) into a form which can be solved analytically on computer. Two particular cases are discussed: 1. \(\dot M\) =C ^te ; 2. \(\dot M\) ÷M ^α (α unspecified).     

Classical Scattering and Block Regularization for the Homogeneous Central Field Problem

The present paper offers an alternative point of view of block regularization for the motion of a particle in a central potential field of the form –x ^–, where x is the distance between the particle and the source and some positive real number.Working in the physical space, we consider the scattering angle determined by the path of the particle as a function of angular momentum. We prove that a particle flow is passing over the collision singularity preserving differentiability with respect to initial data if and only if = 2(1–1/n), n positive integer, n 2.This result coincides with the outcome of block regularization applied by McGehee to the same dynamical problem. We discuss that this identity was to expect since both methods target the same physical constraint over the flow.     

Regularization of Motion Equations with L-Transformation and Numerical Integration of the Regular Equations

The sets of L-matrices of the second, fourth and eighth orders are constructed axiomatically. The defining relations are taken from the regularization of motion equations for Keplerian problem. In particular, the Levi-Civita matrix and KS-matrix are L-matrices of second and fourth order, respectively. A theorem on the ranks of L-transformations of different orders is proved. The notion of L-similarity transformation is introduced, certain sets of L-matrices are constructed, and their classification is given. An application of fourth order L-matrices for N-body problem regularization is given. A method of correction for regular coordinates in the Runge–Kutta–Fehlberg integration method for regular motion equations of a perturbed two-body problem is suggested. Comparison is given for the results of numerical integration in the problem of defining the orbit of a satellite, with and without the above correction method. The comparison is carried out with respect to the number of calls to the subroutine evaluating the perturbational accelerations vector. The results of integration using the correction turn out to be in a favorable position.     

Chaotic behaviour of trajectories for the asteroidal resonances

A systematic study of the main asteroidal resonances of the third and fourth order is performed using mapping techniques. For each resonance one-parameter family of surfaces of section is presented together with a simple energy graph which helps to understand and predict the changes in the surfaces of section within the family. As the truncated Hamiltonian for the planar, elliptic, restricted three-body problem is used for the mapping, the method is expected to fail for high eccentricities. We compared, therefore, the surfaces of section with trajectories calculated by symplectic integrators of the fourth and six order employing the full Hamiltonian. We found a good agreement for small eccentricities but differences for the higher eccentricities (e 0.3).     

Maximum Entropy Regularization Method of Ionospheric Occultation

With the increasing of applications of Global Positioning System (GPS), the research on the factors affecting the radio signals is becoming more and more important. One of the most significant effects on ionosphere monitoring is the ionospheric radio occultation (IRO), and the IRO data is now serving as one of the most important monitoring sources for the ionospheric measurements. Meanwhile the inversion of the occultation data is a popular topic. The traditional Abel inversion process through compensated total electron content (TEC) is a linear inversion method, thus it would transfer the measurement errors directly to the inversion results. In order to improve the occultation results, we introduce in this paper two nonlinear methods, namely the regularization method and the maximum entropy regularization method. Through designed simulative experiments, we verify and compare these three methods, and conclude that the maximum entropy regularization method can reduce significantly the influence of measurement errors.     

电离层掩星反演的正则最大熵法

电离层掩星数据现已成为电离层观测数据的重要来源,对掩星数据的反演研究一直是掩星研究的热点.传统采用的改正TEC(1btal Electron Content)的Abel变换反演法为线性反演法,它会把测量误差带入反演结果中.为改善反演效果受测量误差的影响,引入两种非线性的反演方法一正则化和正则最大熵反演法.随后设计模拟试验,对3种方法进行验证、比较,得到正则最大熵反演法可以很好地减小测量误差的影响.     

Properties of collision trajectories of asteroids with the earth

Close encounters of asteroids with the Earth can lead to a transition to resonant orbits, close encounters, and collisions in the future. The structure of the sets of possible collisions of asteroids with the Earth is similar to a fractal structure thanks to resonant returns of asteroids. Areas of semi-major axes and positions and sizes of gaps leading to collisions with the Earth for Apophis asteroids 2007 VK184 and 2011 AG5 are given.     

Automated design of gravity-assist trajectories to Mars and the outer planets

In this paper, a new approach to planetary mission design is described which automates the search for gravity-assist trajectories. This method finds all conic solutions given a range of launch dates, a range of launch energies and a set of target planets. The new design tool is applied to the problems of finding multiple encounter trajectories to the outer planets and Venus gravity-assist trajectories to Mars. The last four-planet grand tour opportunity (until the year 2153) is identified. It requires an Earth launch in 1996 and encounters Jupiter, Uranus, Neptune, and Pluto. Venus gravity-assist trajectories to Mars for the 30 year period 1995–2024 are examined. It is shown that in many cases these trajectories require less launch energy to reach Mars than direct ballistic trajectories.Assistant Professor, School of Aeronautics and AstronauticsGraduate Student, School of Aeronautics and Astronautics