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.

This library 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

Multiple installation methods are supported:

Logo	Platform	Command
	PyPI	python -m pip install lamberthub
	GitHub	python -m pip install https://github.com/jorgepiloto/lamberthub/archive/main.zip

Available solvers

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

Returns

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.