How to use

Once that you have installed the library, its time for you to learn how to use it. The main goal of lamberthub is very simple: provide a collection of algorithms for solving the Lambert’s problem.

All the routines are implemented in the form of Python functions, who’s name is given by the combination of original author’s name plus the year of publication, that is: authorYYYY.

Cecking for available solvers

To answer this question, simply run the following code snippet or refer to the official package API reference:

from lamberthub import ALL_SOLVERS
print([solver.__name__ for solver in ALL_SOLVERS])
['gauss1809', 'battin1984', 'gooding1990', 'avanzini2008', 'arora2013', 'vallado2013', 'izzo2015']

In addition, lamberthub provides other lists holding algorithms which present particular features such us multi-revolutions or high-robustness. These macros are listed down:

from lamberthub import ALL_SOLVERS, ZERO_REV_SOLVERS, MULTI_REV_SOLVERS, ROBUST_SOLVERS

Import a particular solver

If you are only interested in using a particular solver, you can easily import it by running:

from lamberthub import authorYYYY

where author is the name of the author which developed the solver and YYYY the year of publication. Any of the solvers hosted in the ALL_SOLVERS macro can be used.

If you would like to use a solver which is not defined in lamberthub, open a solver request in the issues board detailing all the information related to the algorithm and any useful reference which can help to implement it.

Input and output parameters

Any of the routines in the library has the same number of input and output parameters, that is because they all solve for the same astrodynamics problem.

As said before, any Lambert’s problem algorithm implemented in lamberthub is a Python function built with the following API architecture:

Parameters

• mu: the gravitational parameter, that is the mass of the attracting body times the gravitational constant.

• r1: initial position vector, needs to be a NumPy array instance.

• r2: final position vector, needs to be a NumPy array instance.

• tof: time of flight between initial and final vectors.

• M: the number of desired revolutions. If zero, direct transfer is assumed.

• prograde: this parameter controls the inclination of the final orbit. If set to True, the transfer will have an inclination between 0 and 90 degrees while if False inclinations between 90 and 180 are provided.

• low_path: selects the type of path when more than two solutions are available. There is no actual advantage on one or another solution, unless you have particular constrains on your mission. If True, the maximum value for the independent variable (when two solutions) is selected.

• 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, it returns additional information such us the number of iterations.

Returns

• v1: initial velocity vector, it is a NumPy array instance.

• v2: final velocity vector, it is a NumPy array instance.

• numiter: number of iterations. Only of full_output has been set to True.

• tpi: time per iteration. Only if full_output has been set to True.

A real example

Let us work over a real example1. Suppose you want to solve for the orbit of an interplanetary vehicle (i.e. Sun is the main attractor) form which you know that the initial and final positions are given by:

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

the dimension of previous vectors is astronomical units [AU] and the time of flight, given in years, is known to be $$\Delta t = 0.010794065 \text{[year]}$$. The orbit is seen to be prograde (inclination is less than $$90^{\circ}$$) and direct $$M=0$$. Remember that when $$M=0$$, there is only one possible solution, so the low_path flag will not play any role in this problem.

To solve for the problem, we first import a solver. Let us use Gooding’s one:

from lamberthub import gooding1990

Next, we specify the initial conditions of the problem:

# Import NumPy for declaring position vectors
import numpy as np

# Initial conditions for the problem
mu_sun = 39.47692641  # [AU ** 3 / year ** 2]
r1 = np.array([0.159321004, 0.579266185, 0.052359607])  # [AU]
r2 = np.array([0.057594337, 0.605750797, 0.068345246])  # [AU]
tof = 0.010794065  # [year]

Finally, we just need to solve for it. Notice that, as explained before, the default value for the prograde flag is True, which matches the one from problem’s statement.

# Solving the problem
v1, v2 = gooding1990(mu_sun, r1, r2, tof)

# Let us print the results
print(f"Initial velocity: {v1} [AU / years]\nFinal velocity: {v2} [AU / years]")
Initial velocity: [-9.303608    3.01862016  1.53636008] [AU / years]
Final velocity: [-9.5111862   1.88884006  1.4213781 ] [AU / years]

previous values are the same ones comming from the original example.

1

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