# lamberthub.ecc_solvers.avanzini

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

 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. _get_eccT_limits(geometry) Computes transverse eccentricity value limits as of problem geometry. _get_eccT_at_x(geometry) Returns proper transverse eccentricity function depending on the value
lamberthub.ecc_solvers.avanzini.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.

Parameters
• mu (float) – Gravitational parameter, equivalent to $$GM$$ of attractor body.

• r1 (numpy.array) – Initial position vector.

• r2 (numpy.array) – Final position vector.

• M (int) – Number of revolutions. Must be equal or greater than 0 value.

• low_path (bool) – If two solutions are available, it selects between high or low path.

• maxiter (int) – Maximum number of iterations.

• atol (float) – Absolute tolerance.

• rtol (float) – Relative tolerance.

• full_output (bool) – If True, the number of iterations and time per iteration are also returned.

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.

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.

lamberthub.ecc_solvers.avanzini._get_eccT_limits(geometry)

Computes transverse eccentricity value limits as of problem geometry.

lamberthub.ecc_solvers.avanzini._get_eccT_at_x(geometry)

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

Parameters
• ecc_H (float) – Lower value limit.

• ecc_P (float) – Upper value limit.

• dtheta (float) – Transfer angle.

Returns

eccT_at_x – A function to be evaluated at independent variable values.

Return type

function

Notes

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