lamberthub: a hub of Lambert's problem solvers

A Python library designed to provide solutions to Lambert's problem, a classical problem in astrodynamics that involves determining the orbit of a spacecraft given two points in space and the time of flight between them. The problem is essential for trajectory planning, particularly for interplanetary missions.

Lamberthub implements multiple algorithms, each named after its author and publication year, for solving different variations of Lambert's problem. These algorithms can handle different types of orbits, including multi-revolution paths and direct transfers.

Installation

Install lamberthub by running:

python -m pip install lamberthub

Available solvers

Algorithm	Reference
gauss1809	C. F. Gauss, Theoria motus corporum coelestium in sectionibus conicis solem ambientium. 1809.
battin1984	R. H. Battin and R. M. Vaughan, “An elegant lambert algorithm,” Journal of Guidance, Control, and Dynamics, vol. 7, no. 6, pp. 662–670, 1984.
gooding1990	R. Gooding, “A procedure for the solution of lambert’s orbital boundary-value problem,” Celestial Mechanics and Dynamical Astronomy, vol. 48, no. 2, pp. 145–165, 1990.
avanzini2008	G. Avanzini, “A simple lambert algorithm,” Journal of Guidance, Control, and Dynamics, vol. 31, no. 6, pp. 1587–1594, 2008.
arora2013	N. Arora and R. P. Russell, “A fast and robust multiple revolution lambert algorithm using a cosine transformation,” Paper AAS, vol. 13, p. 728, 2013.
vallado2013	D. A. Vallado, Fundamentals of astrodynamics and applications. Springer Science & Business Media, 2013, vol. 12.
izzo2015	D. Izzo, “Revisiting lambert’s problem,” Celestial Mechanics and Dynamical Astronomy, vol. 121, no. 1, pp. 1–15, 2015.

Using a solver

Any Lambert's problem algorithm implemented in lamberthub is a Python function which accepts the following parameters:

from lamberthub import authorYYYY


v1, v2 = authorYYYY(
    mu, r1, r2, tof, M=0, prograde=True, low_path=True,  # Type of solution
    maxiter=35, atol=1e-5, rtol=1e-7, full_output=False  # Iteration config
)

where author is the name of the author which developed the solver and YYYY the year of publication. Any of the solvers hosted by the ALL_SOLVERS list.

Parameters and Returns

Parameters	Description
mu	The gravitational parameter, i.e., mass of the attracting body times the gravitational constant. Equivalent to gravitational constant times the mass of the attractor body.
r1	Initial position vector.
r2	Final position vector.
tof	Time of flight between initial and final vectors.
M	The number of revolutions. If zero (default), direct transfer is assumed.
prograde	Controls the inclination of the final orbit. If True, inclination between 0 and 90 degrees. If False, inclination between 90 and 180 degrees.
low_path	Selects the type of path when more than two solutions are available. No specific advantage unless there are mission constraints.
maxiter	Maximum number of iterations allowed when computing the solution.
atol	Absolute tolerance for the iterative method.
rtol	Relative tolerance for the iterative method.
full_output	If True, returns additional information such as the number of iterations.

Returns	Description
v1	Initial velocity vector.
v2	Final velocity vector.
numiter	Number of iterations (only if full_output is True).
tpi	Time per iteration (only if full_output is True).

Examples

Example: solving for a direct and prograde transfer orbit

Problem statement

Suppose you want to solve for the orbit of an interplanetary vehicle (that is Sun is the main attractor) form which you know that the initial and final positions are given by:

$$\vec{r_1} = \begin{bmatrix} 0.159321004 \\ 0.579266185 \\ 0.052359607 \end{bmatrix} \text{ [AU]} \quad \quad \vec{r_2} = \begin{bmatrix} 0.057594337 \\ 0.605750797 \\ 0.068345246 \end{bmatrix} \text{ [AU]} \quad \quad$$

The time of flight is $\Delta t = 0.010794065$ years. The orbit is prograde and direct, thus $M=0$. Remember that when $M=0$, there is only one possible solution, so the low_path flag does not play any role in this problem.

Solution

For this problem, gooding1990 is used. Any other solver would work too. Next, the parameters of the problem are instantiated. Finally, the initial and final velocity vectors are computed.

from lamberthub import gooding1990
import numpy as np


mu_sun = 39.47692641
r1 = np.array([0.159321004, 0.579266185, 0.052359607])
r2 = np.array([0.057594337, 0.605750797, 0.068345246])
tof = 0.010794065

v1, v2 = gooding1990(mu_sun, r1, r2, tof, M=0, prograde=True)
print(f"Initial velocity: {v1} [AU / years]")
print(f"Final velocity:   {v2} [AU / years]")

Result

Initial velocity: [-9.303608    3.01862016  1.53636008] [AU / years]
Final velocity:   [-9.511186    1.88884006  1.42137810] [AU / years]

Directly taken from An Introduction to the Mathematics and Methods of Astrodynamics, revised edition, by R.H. Battin, problem 7-12.