:mod:`lamberthub.ecc_solvers.avanzini`
======================================

.. py:module:: lamberthub.ecc_solvers.avanzini
   :noindex:

.. autoapi-nested-parse::

   Lambert's problem solver using the method proposed by Giulio Avanzini in 2008.

   This module holds the implementation of the algorithm devised by Avanzini, which
   appeared for the first time in his article [1]. The solver takes advantage of
   the conservation of the eccentricity projection along the chord direction, so it
   iterates over this orbital element till the time of transfer from Kepler's
   equation matches the desired transfer one.

   However, the original routines was not developed to work under multi-revolutions
   conditions. In addition, Avanzini did not provide the derivative of Kepler's
   equation with respect to the transverse eccentricity component, forcing a
   bounded numerical method (bisection or regulafalsi). This was solved by Quan He,
   which proposed new changes to Avanzini's original algorithm in an article [2]
   proposed in 2010.

   Finally, a new method [3] was proposed by Changxuan Wen in 2014, simplifying all
   previous routines.

   [1] Avanzini, G. (2008). A simple Lambert algorithm. Journal of guidance,
       control, and dynamics, 31(6), 1587-1594.

   [2] He, Q., Li, J., & Han, C. (2010). Multiple-revolution solutions of the
       transverse-eccentricity-based lambert problem. Journal of guidance,
       control, and dynamics, 33(1), 265-269.

   [3] Wen, C., Zhao, Y., & Shi, P. (2014). Derivative analysis and algorithm
       modification of transverse-eccentricity-based Lambert problem. Journal of
       Guidance, Control, and Dynamics, 37(4), 1195-1201.



Module Contents
---------------


Functions
~~~~~~~~~

.. autoapisummary::


   lamberthub.ecc_solvers.avanzini.avanzini2008


   lamberthub.ecc_solvers.avanzini._get_eccT_limits


   lamberthub.ecc_solvers.avanzini._get_eccT_at_x



.. py:function:: avanzini2008(mu, r1, r2, tof, M=0, prograde=True, low_path=True, maxiter=35, atol=1e-05, rtol=1e-07, full_output=False)

   Solves Lambert problem using Gooding's devised algorithm.

   :param mu: Gravitational parameter, equivalent to :math:`GM` of attractor body.
   :type mu: float
   :param r1: Initial position vector.
   :type r1: numpy.array
   :param r2: Final position vector.
   :type r2: numpy.array
   :param M: Number of revolutions. Must be equal or greater than 0 value.
   :type M: int
   :param prograde: If `True`, specifies prograde motion. Otherwise, retrograde motion is imposed.
   :type prograde: bool
   :param low_path: If two solutions are available, it selects between high or low path.
   :type low_path: bool
   :param maxiter: Maximum number of iterations.
   :type maxiter: int
   :param atol: Absolute tolerance.
   :type atol: float
   :param rtol: Relative tolerance.
   :type rtol: float
   :param full_output: If True, the number of iterations and time per iteration are also returned.
   :type full_output: bool

   :returns: * **v1** (*numpy.array*) -- Initial velocity vector.
             * **v2** (*numpy.array*) -- Final velocity vector.
             * **numiter** (*int*) -- Number of iterations.
             * **tpi** (*float*) -- Time per iteration in seconds.

   .. rubric:: Notes

   The following routine might be simplified making use of private functions.
   However, we decided to expose all the auxiliary routines to properly
   reproduce original report figures.


.. py:function:: _get_eccT_limits(geometry)

   Computes transverse eccentricity value limits as of problem geometry.


.. py:function:: _get_eccT_at_x(geometry)

   Returns proper transverse eccentricity function depending on the value
   of the transfer angle.

   :param ecc_H: Lower value limit.
   :type ecc_H: float
   :param ecc_P: Upper value limit.
   :type ecc_P: float
   :param dtheta: Transfer angle.
   :type dtheta: float

   :returns: **eccT_at_x** -- A function to be evaluated at independent variable values.
   :rtype: function

   .. rubric:: Notes

   These are equations (16) and (18) from the official report [1].