Package org.orekit.control.heuristics.lambert

Class LambertSolver

java.lang.Object
org.orekit.control.heuristics.lambert.LambertSolver
public class LambertSolver extends Object
Lambert position-based Initial Orbit Determination (IOD) algorithm, assuming Keplerian motion. The method is also used for trajectory design, specially in interplanetary missions. This solver combines Dario Izzo's algorithm with Gim Der design to find all possible solutions.

An orbit is determined from two position vectors. References: Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education, 1999. Lancaster, E.R. and Blanchard, R.C., A Unified Form of Lambert’s Theorem, Goddard Space Flight Center, 1968. Dario Izzo. Revisiting Lambert’s problem. Celestial Mechanics and Dynamical Astronomy, 2015. https://arxiv.org/abs/1403.2705 Gim J. Der. The Superior Lambert Algorithm. Advanced Maui Optical and Space Surveillance Technologies, 2011. https://amostech.com/TechnicalPapers/2011/Poster/DER.pdf

Since:
14.0
Author:
Joris Olympio, Romain Serra, Rafael Ayala

  • Constructor Details

    • LambertSolver

      public LambertSolver(double mu, HouseholderParameters householderParameters)
      Constructor from Householder parameters object.
      Parameters:
      mu - gravitational constant
      householderParameters - parameters for the Householder solver

    • LambertSolver

      public LambertSolver(double mu, int householderMaxIterations, double householderAtol, double householderRtol)
      Constructor from direct Householder parameter values.
      Parameters:
      mu - gravitational constant
      householderMaxIterations - maximum number of iterations for the Householder solver
      householderAtol - absolute tolerance for the Householder solver
      householderRtol - relative tolerance for the Householder solver

    • LambertSolver

      public LambertSolver(double mu)
      Constructor with default Householder solver parameters.
      Parameters:
      mu - gravitational constant

  • Method Details

    • solve

      public List<LambertSolution> solve(boolean posigrade, LambertBoundaryConditions boundaryConditions)
      Lambert's solver.
      Parameters:
      posigrade - flag indicating the direction of motion
      boundaryConditions - LambertBoundaryConditions holding the boundary conditions
      Returns:
      a list of solutions

    • solve

      public List<LambertSolution> solve(boolean posigrade, int nRevs, LambertBoundaryConditions boundaryConditions)
      Lambert's solver, with user provided number of complete revolutions.
      Parameters:
      posigrade - flag indicating the direction of motion
      nRevs - number of complete revolutions
      boundaryConditions - LambertBoundaryConditions holding the boundary conditions
      Returns:
      a list of solutions

    • initialSetup

      public void initialSetup(boolean posigrade, LambertBoundaryConditions boundaryConditions)
      Initial set up of geometry and auxiliary variables.
      Parameters:
      posigrade - flag indicating the direction of motion
      boundaryConditions - LambertBoundaryConditions holding the boundary conditions

    • initialGuessX

      public static double initialGuessX(double tau, double sigma, int nRevs, boolean lowPath)
      Initial guess for x0.
      Parameters:
      tau - value of tau
      sigma - value of sigma
      nRevs - number of complete revolutions
      lowPath - flag indicating low path
      Returns:
      initial guess for x0

    • calculateY

      public static double calculateY(double x, double sigma)
      Calculate value of y from x (and sigma).
      Parameters:
      x - value of x
      sigma - value of sigma
      Returns:
      value of y

    • householderSolver

      public static double householderSolver(double x0, double tau0, double sigma, int nRevs, int maxIterations, double atol, double rtol)
      Householder solver for the Lambert problem.
      Parameters:
      x0 - initial guess for x
      tau0 - value of tau0
      sigma - value of sigma
      nRevs - number of complete revolutions
      maxIterations - maximum number of iterations
      atol - absolute tolerance for convergence
      rtol - relative tolerance for convergence
      Returns:
      value of x

    • reconstructVrVt

      public static double[] reconstructVrVt(double x, double y, double r1, double r2, double sigma, double gamma, double rho, double zeta)
      Reconstruct the values of Vr and Vt (radial and transversal components of velocity). These are used together with the ir and it vectors to compute the velocity vectors at the beginning and end of the transfer.
      Parameters:
      x - value of x
      y - value of y
      r1 - value of r1
      r2 - value of r2
      sigma - value of sigma
      gamma - value of gamma
      rho - value of rho
      zeta - value of zeta
      Returns:
      an array containing the values of Vr and Vt at the begginning and end of the transfer

    • hyperg2F1

      public static double hyperg2F1(double a, double b, double c, double z, double eps, int maxIter)
      Calculate the value of Gaussian hypergeometric function 2F1. Currently we use the raw series expansion. This means we have the following constraints: |z| smaller than 1, c larger than 0, c != 0. Implementation based on Taylor series expansion method (a) in John Pearson's thesis https://people.maths.ox.ac.uk/porterm/research/pearson_final.pdf , page 31.
      Parameters:
      a - value of a
      b - value of b
      c - value of c
      z - value of z (|z| smaller than 1)
      eps - convergence threshold
      maxIter - maximum number of iterations
      Returns:
      value of the 2F1 hypergeometric function

    • computeJacobian

      public RealMatrix computeJacobian(LambertBoundaryConditions boundaryConditions, LambertBoundaryVelocities velocities)
      Computes the Jacobian matrix of the Lambert solution. The rows represent the initial and terminal velocity vectors. The columns represent the parameters: initial time, initial position, terminal time, terminal velocity.

      Reference: Di Lizia, P., Armellin, R., Zazzera, F. B., and Berz, M. High Order Expansion of the Solution of Two-Point Boundary Value Problems using Differential Algebra: Applications to Spacecraft Dynamics.

      Parameters:
      boundaryConditions - Lambert boundary conditions
      velocities - velocities of a Lambert solution to compute the Jacobian for
      Returns:
      Jacobian matrix

    • getMu

      public double getMu()