boinor.core.iod

module containing functions related to Initial Orbit Determination (IOD) in the core sub-package

Functions

vallado(k, r0, r, tof, M, prograde, lowpath, numiter, rtol)

Solves the Lambert's problem.

izzo(k, r1, r2, tof, M, prograde, lowpath, numiter, rtol)

Aplies izzo algorithm to solve Lambert's problem.

Module Contents

boinor.core.iod.vallado(k, r0, r, tof, M, prograde, lowpath, numiter, rtol)

Solves the Lambert’s problem.

The algorithm returns the initial velocity vector and the final one, these are computed by the following expresions:

\[\begin{split}\vec{v_{o}} &= \frac{1}{g}(\vec{r} - f\vec{r_{0}}) \\ \vec{v} &= \frac{1}{g}(\dot{g}\vec{r} - \vec{r_{0}})\end{split}\]

Therefore, the lagrange coefficients need to be computed. For the case of Lambert’s problem, they can be expressed by terms of the initial and final vector:

\[\begin{split}\begin{aligned} f = 1 -\frac{y}{r_{o}} \\ g = A\sqrt{\frac{y}{\mu}} \\ \dot{g} = 1 - \frac{y}{r} \\ \end{aligned}\end{split}\]

Where y(z) is a function that depends on the boinor.core.stumpff coefficients:

\[\begin{split}y = r_{o} + r + A\frac{zS(z)-1}{\sqrt{C(z)}} \\ A = \sin{(\Delta \nu)}\sqrt{\frac{rr_{o}}{1 - \cos{(\Delta \nu)}}}\end{split}\]

The value of z to evaluate the stump functions is solved by applying a Numerical method to the following equation:

\[z_{i+1} = z_{i} - \frac{F(z_{i})}{F{}'(z_{i})}\]

Function F(z) to the expression:

\[F(z) = \left [\frac{y(z)}{C(z)} \right ]^{\frac{3}{2}}S(z) + A\sqrt{y(z)} - \sqrt{\mu}\Delta t\]
Parameters:

  • k (float) – Gravitational Parameter

  • r0 (numpy.ndarray) – Initial position vector

  • r (numpy.ndarray) – Final position vector

  • tof (float) – Time of flight

  • M (int) – Number of revolutions

  • prograde (boolean) – Controls the desired inclination of the transfer orbit.

  • lowpath (boolean) – If True or False, gets the transfer orbit whose vacant focus is below or above the chord line, respectively.

  • numiter (int) – Number of iterations to

  • rtol (int) – Number of revolutions

Returns:

  • v0 (numpy.ndarray) – Initial velocity vector

  • v (numpy.ndarray) – Final velocity vector

Examples

>>> from boinor.core.iod import vallado
>>> from astropy import units as u
>>> import numpy as np
>>> from boinor.bodies import Earth
>>> k = Earth.k.to(u.km ** 3 / u.s ** 2)
>>> r1 = np.array([5000, 10000, 2100]) * u.km # Initial position vector
>>> r2 = np.array([-14600, 2500, 7000]) * u.km # Final position vector
>>> tof = 3600 * u.s # Time of flight
>>> v1, v2 = vallado(k.value, r1.value, r2.value, tof.value, M=0, prograde=True, lowpath=True, numiter=35, rtol=1e-8)
>>> v1 = v1 * u.km / u.s
>>> v2 = v2 * u.km / u.s
>>> print(v1, v2)
[-5.99249503  1.92536671  3.24563805] km / s [-3.31245851 -4.19661901 -0.38528906] km / s

Notes

This procedure can be found in section 5.3 of Curtis [Cur13], with all the theoretical description of the problem. Analytical example can be found in the same book under name Example 5.2.

boinor.core.iod.izzo(k, r1, r2, tof, M, prograde, lowpath, numiter, rtol)

Aplies izzo algorithm to solve Lambert’s problem.

Parameters:

  • k (float) – Gravitational Constant

  • r1 (numpy.ndarray) – Initial position vector

  • r2 (numpy.ndarray) – Final position vector

  • tof (float) – Time of flight between both positions

  • M (int) – Number of revolutions

  • prograde (boolean) – Controls the desired inclination of the transfer orbit.

  • lowpath (boolean) – If True or False, gets the transfer orbit whose vacant focus is below or above the chord line, respectively.

  • numiter (int) – Number of iterations

  • rtol (float) – Error tolerance

Returns:

  • v1 (numpy.ndarray) – Initial velocity vector

  • v2 (numpy.ndarray) – Final velocity vector