On the solution to every Lambert problem

Russell, Ryan P.

doi:10.1007/s10569-019-9927-z

Cited by 17 publications

(4 citation statements)

References 29 publications

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“…With ρ 1 and ρ n , denoting the observed distances at moments r 1 and r n , respectively, and R i representing the position vector of the observing station, if we know ρ 1 and ρ n , we can get two position vectors r 1 and r n based on r i = R i + ρ i L i (i = 1, n), so that the purely angular IOD problem is converted into an orbit-determination problem based on two position vectors, which is the so-called IOD Lambert problem (the orbital parameters are calculated using the position vectors of the two moments). The Lambert problem is well known in dynamical astronomy, celestial mechanics, and astrodynamics communities for objects governed by Keplerian dynamics [19]. The classical methods can be used to solve the standard Lambert problem given two positions, such as the Gibbs method or the Herrick-Gibbs method [20,21].…”

Section: The Improvement Of the Rs Iod Methods And The Association Me...mentioning

confidence: 99%

An Improved Range-Searching Initial Orbit-Determination Method and Correlation of Optical Observations for Space Debris

Lei,

Xia,

Liu

et al. 2023

Applied Sciences

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The Changchun Observatory of the National Astronomical Observatories, Chinese Academy of Sciences, and the Shanghai Astronomical Observatory are used to generate very short arc (VSA) angle observations of objects in low Earth orbit (LEO) and geostationary orbit (GEO) with their ground-based electrical–optical telescope arrays (EA), the Changchun EA and SAO FocusGEO, respectively. These observations are used in this paper. The range-searching (RS) algorithm for initial orbit determination (IOD) is improved through the multiple combinations of observations and the dynamic range-searching step length. Two different computation modes (the normal mode and the refining mode) of the IOD computation process are proposed. The geometrical method for the association is used. The IOD and association methods are extended to the real optical observations for both LEO and GEO objects. The results show that the IOD success rate of arcs from the LEO objects is about 91%, the error of the semimajor axis (SMA) of the initial orbital elements is less than 50 km, and the correlation accuracy rate is about 89%. The IOD success rate of arcs from the GEO objects is higher than 88%, and the correlation accuracy rate is greater than 87%. The recent COSMOS 1408 antisatellite test (ASAT) generated a large amount of debris. The algorithm of this paper and the observations of Changchun EA are used to initially identify new debris, possibly from the ASAT through initial orbit determination and track association. Finally, 64 suspected new pieces of debris can be found. The results show the effectiveness of the IOD and the correlation algorithm, as well as the potential application of the optical–electrical array in studying space events.

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Section: The Improvement Of the Rs Iod Methods And The Association Me...mentioning

confidence: 99%

An Improved Range-Searching Initial Orbit-Determination Method and Correlation of Optical Observations for Space Debris

Lei,

Xia,

Liu

et al. 2023

Applied Sciences

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“…The output is only the initial velocity v0 as the terminal velocity can be obtained through orbital propagation once the initial velocity is solved. This type of sample form is defined as (5) where the subscript 0 and f denotes the start and end of the transfer trajectory, respectively. Thus, when trained with sample form in Eq.…”

Section: A Definition Of Sample Form and Featuresmentioning

confidence: 99%

“…Since the perturbation due to the J2 zonal harmonics has the most significant effect around all planets in the solar system, a body of research exists that addressed the problem of solving the perturbed Lambert problem accounting for the J2 effect [3], [4]. This body of research can be classified into two categories: indirect methods and shooting methods [5]. Indirect methods transform the perturbed Lambert problem into the solution of a system of parametric nonlinear algebraic equations.…”

Section: Iintroductionmentioning

confidence: 99%

Fast Solver for J2-Perturbed Lambert Problem Using Deep Neural Network

Yang¹,

Li²,

Feng³

et al. 2022

Journal of Guidance, Control, and Dynamics

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This paper presents a novel and fast solver for the J2-perturbed Lambert problem. The solver consists of an intelligent initial guess generator combined with a differential correction procedure. The intelligent initial guess generator is a deep neural network that is trained to correct the initial velocity vector coming from the solution of the unperturbed Lambert problem. The differential correction module takes the initial guess and uses a forward shooting procedure to further update the initial velocity and exactly meet the terminal conditions. Eight sample forms are analyzed and compared to find the optimum form to train the neural network on the J2-perturbed Lambert problem. The accuracy and performance of this novel approach will be demonstrated on a representative test case: the solution of a multi-revolution J2-perturbed Lambert problem in the Jupiter system. We will compare the performance of the proposed approach against a classical standard shooting

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“…The weights were selected to emphasize the minimization of v ∞,2 since this influences the magnitude of the orbit insertion ∆v. The ivLam solver, a fast Lambert problem solver, developed by the University of Texas [56], was used to design the heliocentric cruise from a given design gene. The integer programming problem in Equation ( 3) was solved using the proprietary GA optimizer of MATLAB [25].…”

Section: Heliocentric Cruisementioning

confidence: 99%

Trajectory Design of Perseus: A CubeSat Mission Concept to Phobos

et al. 2020

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The Martian satellites Phobos and Deimos hold many unanswered questions that may provide clues to the origin of Mars. These moons are low Δv stopover sites to Mars. Some human missions to Mars typically identify Phobos and Deimos as staging bases for Mars surface exploration. Astronauts could base initial operations there in lieu of repeated voyages to and from the planet surface, to refuel transiting spacecraft, to teleoperate robotics and other critical machinery, and to develop habitable infrastructure ahead of human landings. Despite their strategic and scientific significance, there has been no successful dedicated mission to either moon. For this reason, we propose Perseus, a geological imaging CubeSat mission to Phobos. Perseus, a 27U, 54kg CubeSat will return thermal and visible images at resolutions better than currently available over most of Phobos’ surface. This includes visible images at 5m/pixel and thermal images at 25m/pixel of Phobos’ surface. The Perseus mission is nominally intended to be a co-orbital mission, where the spacecraft will encounter Phobos on its Martian orbit. However, a hyperbolic rendezvous mission concept, to image Phobos on a hyperbolic flyby, is also considered to reduce the risks associated with orbit capture and to reduce mission costs. This paper presents the preliminary feasibility, science objectives, and technological development challenges of achieving these science goals. We then formulate two rendezvous concepts as a series of three nonlinear optimization problems that span the design tree of mission concepts. The tree’s root node is the heliocentric cruise problem, which identifies the near-optimal launch and arrival windows for the Perseus spacecraft. The leaf nodes of the design tree are the two rendezvous concepts that identify near-optimal co-orbital and hyperbolic trajectories for Phobos’ reconnaissance. The design problems are solved using evolutionary algorithms, and the performance of the selected mission concepts is then examined. The results indicate that a co-orbital encounter allows about one encounter per day with about 6 min per encounter. The hyperbolic encounter, on the other hand, allows a single encounter where the spacecraft will spend about 2 min in the imaging region with respect to Phobos. The spacecraft will obtain higher resolution images of Phobos on this feasible region than have ever been seen for most of the surface. These detailed images will help identify candidate landing sites and provide critical data to derisk future surface missions to Phobos.

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On the solution to every Lambert problem

Cited by 17 publications

References 29 publications

An Improved Range-Searching Initial Orbit-Determination Method and Correlation of Optical Observations for Space Debris

An Improved Range-Searching Initial Orbit-Determination Method and Correlation of Optical Observations for Space Debris

Fast Solver for J2-Perturbed Lambert Problem Using Deep Neural Network

Trajectory Design of Perseus: A CubeSat Mission Concept to Phobos

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