Unified Lambert Tool for Massively Parallel Applications in Space Situational Awareness

Abstract

This paper introduces a parallel-compiled tool that combines several of our recently developed methods for solving the perturbed Lambert problem using modified Chebyshev-Picard iteration. This tool (unified Lambert tool) consists of four individual algorithms, each of which is unique and better suited for solving a particular type of orbit transfer. The first is a Keplerian Lambert solver, which is used to provide a good initial guess (warm start) for solving the perturbed problem. It is also used to determine the appropriate algorithm to call for solving the perturbed problem. The arc length or true anomaly angle spanned by the transfer trajectory is the parameter that governs the automated selection of the appropriate perturbed algorithm, and is based on the respective algorithm convergence characteristics. The second algorithm solves the perturbed Lambert problem using the modified Chebyshev-Picard iteration two-point boundary value solver. This algorithm does not require a Newtonlike shooting method and is the most efficient of the perturbed solvers presented herein, however the domain of convergence is limited to about a third of an orbit and is dependent on eccentricity. The third algorithm extends the domain of convergence of the modified Chebyshev-Picard iteration two-point boundary value solver to about 90% of an orbit, through regularization with the Kustaanheimo-Stiefel transformation. This is the second most efficient of the perturbed set of algorithms. The fourth algorithm uses the method of particular solutions and the modified ChebyshevPicard iteration initial value solver for solving multiple revolution perturbed transfers. This method does require "shooting" but differs from Newton-like shooting methods in that it does not require propagation of a state transition matrix. The unified Lambert tool makes use of the General Mission Analysis Tool and we use it to compute thousands of perturbed Lambert trajectories in parallel on the Space Situational Awareness computer cluster at the LASR Lab, Texas A&M University. We demonstrate the power of our tool by solving a highly parallel example problem, that is the generation of extremal field maps for optimal spacecraft rendezvous (and eventual orbit debris removal). In addition we demonstrate the need for including perturbative effects in simulations for satellite tracking or data association. The unified Lambert tool is ideal for but not limited to space situational awareness applications.

GPU-based high-precision orbital propagation of large sets of initial conditions through Picard–Chebyshev augmentation

“…Later, Koblick and Shankar [15] extended the analysis to the propagation of accurate orbits testing difference force models with NASA's Java Astrodynamics toolkit. Woollands et al [16][17][18] applied the method as numerical integrator for the solution of the Lambert two-point boundary value problem, assessing also the benefits of adopting the Kuustanheimo-Stiefel formulation of the dynamics and proposing a solution for the multi-revolution trajectory design. Swenson et al [19] applied the modified PC method on the circular restricted three-body problem, using the differential correction approach.…”

Kustaanheimo–Stiefel Variables for Planetary Protection Compliance Analysis

“…specified angular momentum vector, and applied it to the design of low thrust trajectories. Woollands et al [55] developed a Lambert solver based on KS coordinates and used it to provide a good initial guess to the Picard-Chebyshev numerical integration of the perturbed two-body problem. Sellamuthu and Sharma analysed the J 2 , J 3 , J 4 terms of Earth's oblateness and the third body luni-solar perturbation when approximated with a Legendre polynomial expansion with KS coordinates [56][57][58].…”

Optimal Midcourse Guidance Algorithm for Exoatmospheric Interception Using Analytical Gradients

“…This kind of problem is abbreviated as the minimum velocity increment problem (MVIP) in this paper. It is worth noting that the aforementioned algorithms cannot be directly applied to the MVIP, because the TOF should be regarded as an unspecified parameter in the optimization process [30][31][32]. Commonly, an additional single-variable algorithm needs to be developed to repeatedly solve Lambert's problem so as to determine the optimal TOF.…”