Utilities Subpackage

latom.utils.coc	@authors: Alberto FOSSA’ Giuliana Elena MICELI
latom.utils.const	@authors: Alberto FOSSA’ Giuliana Elena MICELI
latom.utils.keplerian_orbit	@authors: Alberto FOSSA’ Giuliana Elena MICELI
latom.utils.pickle_utils	@authors: Alberto FOSSA’ Giuliana Elena MICELI
latom.utils.primary	@authors: Alberto FOSSA’ Giuliana Elena MICELI
latom.utils.spacecraft	@authors: Alberto FOSSA’ Giuliana Elena MICELI

Documentation

@authors: Alberto FOSSA’ Giuliana Elena MICELI

class latom.utils.keplerian_orbit.TwoDimOrb(gm, **kwargs)

Bases: object

Defines a two-dimensional orbit from its keplerian parameters.

Parameters

gm (float) – Central body standard gravitational parameter [m^3/s^2]

Other Parameters

• T (float) – Orbital period [s]

• a (float) – Semi-major axis [m]

• e (float) – Eccentricity [-]

• ra (float) – Apoapsis radius [m]

• rp (float) – Periapsis radius [m]

Variables

• GM (float, optional) – Central body standard gravitational parameter [m^3/s^2]

• T (float) – Orbit period [s]

• a (float) – Semi-major axis [m]

• e (float) – Eccentricity [-]

• ra (float) – Apoapsis radius [m]

• rp (float) – Periapsis radius [m]

• va (float) – Velocity at the apoapsis [m/s]

• vp (float) – Velocity at the periapsis [m/s]

• h (float) – Specific angular momentum magnitude [m^2/s]

• energy (float) – Specific energy of the orbit in 2BP [m^2/s^2]

static coe2polar(gm, ta, **kwargs)

Transforms the classical orbital elements into polar coordinates.

Parameters

• gm (float) – Central body standard gravitational parameter [m^3/s^2]

• ta (float or ndarray) – True anomaly [rad]

Other Parameters

• a (float) – Semi-major axis [m]

• e (float) – Eccentricity [-]

• h (float) – Specific angular momentum magnitude [m^2/s]

Returns

• r (float or ndarray) – Position in polar coordinates [m]

• u (float or ndarray) – radial velocity [m/s]

• v (float or ndarray) – tangential velocity [m/s]

static polar2coe(gm, r, u, v)

Transforms the polar coordinates into classical orbital elements.

Parameters

• gm (float) – Central body standard gravitational parameter [m^3/s^2]

• r (float or ndarray) – Position in polar coordinates [m]

• u (float or ndarray) – radial velocity [m/s]

• v (float or ndarray) – tangential velocity [m/s]

Returns

• a (float) – Orbit semi-major axis [m]

• e (float) – Orbit eccentricity [-]

• h (float) – Orbit specific angular momentum magnitude [m^2/s]

• ta (float or ndarray) – True anomaly [rad]

static polar2energy(gm, r, u, v)

Returns the energy given the polar coordinates.

Parameters

• gm (float) – Central body standard gravitational parameter [m^3/s^2]

• r (float or ndarray) – Position in polar coordinates [m]

• u (float or ndarray) – radial velocity [m/s]

• v (float or ndarray) – tangential velocity [m/s]

Returns

(u**2 + v**2)*0.5 - gm/r – Specific energy of the orbit in 2BP [m^2/s^2]

Return type

float or ndarray

static propagate(gm, a, e, tai, taf, nb, tp=0.0)

Propagates the orbit states in polar coordinates for the orbital period duration.

Parameters

• gm (float) – Central body standard gravitational parameter [m^3/s^2]

• a (float) – Orbit semi-major axis [m]

• e (float) – Orbit eccentricity [-]

• tai (float) – True anomaly initial angle [rad]

• taf (float) – True anomaly final angle [rad]

• nb (int) – number of point where the True Anomaly is calculated [-]

• tp (float) – Time at periapsis passage [s]

Returns

• t (ndarray) – Time vector [s]

• states (ndarray) – List of the states values propagated in time [m, m/s, m/s]

class latom.utils.keplerian_orbit.KepOrb(a, e, i, raan, w, ta, gm)

Bases: object

KepOrb defines a Keplerian Orbit.

Parameters

• a (float) – Semi-major axis [m]

• e (float) – Eccentricity [-]

• i (float) – Inclination [rad]

• raan (float) – Right Ascension of the Ascending Node [rad]

• w (float) – Argument of Periapsis [rad]

• ta (float) – True anomaly [rad]

• gm (float) – Central body standard gravitational parameter [m^3/s^2]

Variables

• eps (float) – Smallest number such that 1.0 + eps != 1.0

• a (float) – Semi-major axis [m]

• e (float) – Eccentricity [-]

• i (float) – Inclination [rad]

• W (float) – Right Ascension of the Ascending Node [rad]

• w (float) – Argument of Periapsis [rad]

• ta (float) – True anomaly [rad]

• GM (float, optional) – Central body standard gravitational parameter [m^3/s^2]

• n (float) – Mean motion [rad/s]

• R (ndarray) – Position vector [m]

• V (ndarray) – Velocity vector [m/s]

• H (ndarray) – Specific angular momentum vector [m^2/s]

• E (ndarray) – Eccentricity vector [-]

• h (float) – Specific angular momentum magnitude [m^2/s]

• r (float) – Position vector magnitude [m]

• vr (float) – Radial velocity [m/s]

set_state_vector(r, v)

Set the spacecraft state vector and compute the corresponding COE, H, E.

Parameters

• r (ndarray) – Position vector [m]

• v (ndarray) – Velocity vector [m/s]

set_classical_orbital_elements(a, e, i, raan, w, ta)

Set the spacecraft COE and compute the corresponding state vector, H, E.

Parameters

• a (float) – Semi-major axis [km]

• e (float) – Eccentricity [-]

• i (float) – Inclination [rad]

• raan (float) – Right Ascension of the Ascending Node [rad]

• w (float) – Argument of Periapsis [rad]

• ta (float) – True anomaly [rad]

set_true_anomaly(ta)

Set the spacecraft true anomaly and update the corresponding state vector.

Parameters

ta (float) – True anomaly [rad]

compute_mean_motion()

Computes the spacecraft mean motion.

compute_angular_momentum()

Computes the specific angular momentum vector.

compute_eccentricity()

Compute the spacecraft eccentricity vector.

compute_classical_orbital_elements()

Computes the spacecraft classical orbital elements, specific angular momentum vector and eccentricity vector from its state vector.

compute_state_vector()

Computes the spacecraft state vector, specific angular momentum vector and eccentricity vector from its COE.

compute_eccentric_anomaly(ta)

Compute the eccentric anomaly from a given true anomaly.

Parameters

ta (float) – True anomaly [rad]

Returns

ea – Eccentric anomaly [rad]

Return type

float

compute_true_anomaly(ea)

Compute the true anomaly from a given eccentric anomaly.

Parameters

ea (ndarray or float) – Eccentric anomaly [-]

Returns

ta – True anomaly [rad]

Return type

ndarray or float

compute_periapsis_passage(ta, t)

Compute the time at periapsis passage given the current time and true anomaly.

Parameters

• t (float) – Time [s]

• ta (float) – True anomaly [rad]

Returns

tp – Time at periapsis passage [s]

Return type

float

propagate(ta, t_vec, mode)

Propagate the orbit forward or backward in time solving the Kepler’s time of flight equation.

Parameters

• ta (float) – Initial true anomaly [rad]

• t_vec (ndarray) – Time vector [s]

• mode (str) – fwd for forward propagation or back for backward propagation

Returns

• r_vec (ndarray) – Position vector time series [m]

• v_vec (ndarray) – Velocity vector time series [m/s]

static kepler_eqn(ea, e, n, t, tp)

Kepler’s Time of Flight equation.

Parameters

• ea (float) – Eccentric anomaly [rad]

• e (float) – Eccentricity [-]

• n (float) – Mean motion [rad/s]

• t (float) – Time [s]

• tp (float) – Time at periapsis passage [s]

Returns

f – Current value of Kepler’s equation written as E - e*sin(E) - Me = 0

Return type

float

static kepler_eqn_prime(ea, e, n, t, tp)

First derivative of Kepler’s time of flight equation wrt eccentric anomaly.

Parameters

• ea (float) – Eccentric anomaly [rad]

• e (float) – Eccentricity [-]

• n (float) – Mean motion [rad/s]

• t (float) – Time [s]

• tp (float) – Time at periapsis passage [s]

Returns

fprime – First derivative of Kepler’s equation wrt eccentric anomaly

Return type

float

static kepler_eqn_second(ea, e, n, t, tp)

Second derivative of Kepler’s time of flight equation wrt eccentric anomaly.

Parameters

• ea (float) – Eccentric anomaly [rad]

• e (float) – Eccentricity [-]

• n (float) – Mean motion [rad/s]

• t (float) – Time [s]

• tp (float) – Time at periapsis passage [s]

Returns

fsecond – Second derivative of Kepler’s equation wrt eccentric anomaly

Return type

float

@authors: Alberto FOSSA’ Giuliana Elena MICELI

class latom.utils.spacecraft.Spacecraft(isp, twr, throttle_min=0.0, m0=1.0, m_dry=None, g=9.80665)

Bases: object

Spacecraft class defines the spacecraft characteristics.

Parameters

• isp (float) – Specific impulse [s]

• twr (float) – Thrust over initial weight ratio [-]

• throttle_min (float, optional) – Minimum throttle level [-]. Default is 0.0

• m0 (float, optional) – Initial mass [kg]. Default is 1.0

• m_dry (float or None, optional) – Dry mass [kg]. Default is None for which m_dry is set equal to m0/100

• g (float, optional) – Central body surface gravity [m/s^2]. Default is g0

Variables

• Isp (float) – Specific impulse [s]

• twr (float) – Thrust over initial weight ratio [-]

• m0 (float) – Initial mass [kg]

• m_dry (float) – Dry mass [kg]

• w (float) – Exhaust velocity [m/s]

• T_max (float) – Maximum thrust [N]

• T_min (float) – Minimum thrust [N]

update_twr(twr)

Updates the boundaries value for the thrust throttle

Parameters

twr (float) – Thrust over initial weight ratio [-]

class latom.utils.spacecraft.ImpulsiveBurn(sc, dv)

Bases: object

ImpulsiveBurn class describes an impulsive burn.

Parameters

• sc (Spacecraft) – Instant of Spacecraft class

• dv (float) – Change in velocity corresponding to the impulsive burn [m/s]

Variables

• sc (Spacecraft) – Instant of Spacecraft class

• dv (float) – Change in velocity corresponding to the impulsive burn [m/s]

• mf (float) – Spacecraft final mass after the impulsive burn [kg]

• dm (float) – Propellant mass required for the impulsive burn [kg]

static tsiolkovsky_mf(m0, dv, isp)

Computes the final spacecraft mass for a given velocity change using the Tsiolkovsky rocket equation.

Parameters

• m0 (float) – Initial spacecraft mass [kg]

• dv (float) – Change in velocity [m/s]

• isp (float) – Specific impulse of the spacecraft rocket engine [s]

Returns

mf – Final spacecraft mass [kg]

Return type

float

static tsiolkovsky_dv(m0, mf, isp)

Computes the velocity change for a given initial and final spacecraft masses using the Tsiolkovsky rocket equation.

Parameters

• m0 (float) – Initial spacecraft mass [kg]

• mf (float) – Final spacecraft mass [kg]

• isp (float) – Specific impulse of the spacecraft rocket engine [s]

Returns

dv – Change in velocity [m/s]

Return type

float

@authors: Alberto FOSSA’ Giuliana Elena MICELI

latom.utils.coc.rot1(t)

Elementary rotation matrix around X axis.

Parameters

t (float) – Rotation angle [rad]

Returns

r – 3x3 rotation matrix around X axis

Return type

ndarray

latom.utils.coc.rot3(t)

Elementary rotation matrix around Z axis.

Parameters

t (float) – Rotation angle [rad]

Returns

r – 3x3 rotation matrix around Z axis

Return type

ndarray

latom.utils.coc.eq2per(raan, i, w)

Rotation matrix from inertial, body-centred equatorial reference frame to perifocal reference frame.

Parameters

• raan (float) – Right Ascension of the Ascending Node [rad]

• i (float) – Inclination [rad]

• w (float) – Argument of Periapsis [rad]

Returns

q – 3x3 rotation matrix from equatorial to perifocal reference frames

Return type

ndarray

latom.utils.coc.per2eq(raan, i, w)

Rotation matrix from perifocal reference frame to inertial, body-centred equatorial reference frame.

Parameters

• raan (float) – Right Ascension of the Ascending Node [rad]

• i (float) – Inclination [rad]

• w (float) – Argument of Periapsis [rad]

Returns

q – 3x3 rotation matrix from perifocal to equatorial reference frames

Return type

ndarray

latom.utils.coc.coe2sv_vec(a, e, i, raan, w, ta, gm)

Change of Coordinates from Classical Orbital Elements to State Vector in body-centred inertial reference frame.

Parameters

• a (float) – Semi-major axis [m]

• e (float) – Eccentricity [-]

• i (float) – Inclination [rad]

• raan (float) – Right Ascension of the Ascending Node [rad]

• w (float) – Argument of Periapsis [rad]

• ta (ndarray) – True anomalies [rad]

• gm (float) – Central body standard gravitational parameter [m^3/s^2]

Returns

• r_vec (ndarray) – Position vector [m]

• v_vec (ndarray) – Velocity vector [m/s]

latom.utils.coc.polar2per_vec(r, theta, u, v)

Transformation from polar coordinates to perifocal reference frame.

Parameters

• r (ndarray) – Radius time series [m]

• theta (ndarray) – Angle time series [rad]

• u (ndarray) – Radial velocity time series [m/s]

• v (ndarray) – Tangential velocity time series [m/s]

Returns

• xp (ndarray) – Position along x axis [m]

• yp (ndarray) – Position along y axis [m]

• vxp (ndarray) – Velocity component along x axis [m/s]

• vyp (ndarray) – Velocity component along y axis [m/s]

latom.utils.coc.per2eq_vec(xp, yp, vxp, vyp, raan, i, w)

Transformation from perifocal reference frame to inertial, body-centred equatorial reference frame.

Parameters

• xp (ndarray) – Position along x axis [m]

• yp (ndarray) – Position along y axis [m]

• vxp (ndarray) – Velocity component along x axis [m/s]

• vyp (ndarray) – Velocity component along y axis [m/s]

• raan (float) – Right Ascension of the Ascending Node [rad]

• i (float) – Inclination [rad]

• w (float) – Argument of Periapsis [rad]

Returns

• r_vec (ndarray) – Position vector [m]

• v_vec (ndarray) – Velocity vector [m/s]

latom.utils.coc.polar2eq_vec(r, theta, u, v, raan, i, w)

Transformation from polar coordinates to inertial, body-centred equatorial reference frame.

Parameters

• r (nd array) – Radius time series [m]

• theta (ndarray) – Angle time series [rad]

• u (nd array) – Radial velocity time series [m/s]

• v (ndarray) – Tangential velocity time series [m/s]

• raan (float) – Right Ascension of the Ascending Node [rad]

• i (float) – Inclination [rad]

• w (float) – Argument of Periapsis [rad]

Returns

• r_vec (ndarray) – Position vector [m]

• v_vec (ndarray) – Velocity vector [m/s]

latom.utils.coc.lat_long2cartesian(lat, long, r)

Transformation from latitude, longitude to cartesian coordinates.

Parameters

• lat (float) – Latitude [deg]

• long (float) – Longitude [deg]

• r (float) – Planet radius [m]

@authors: Alberto FOSSA’ Giuliana Elena MICELI

Defines the methods to save and load files using the pickle library

latom.utils.pickle_utils.save(obj, filename)

Save an object.

Parameters

• obj (object) – Object to be serialized

• filename (str) – Full path to file where the object is serialized

latom.utils.pickle_utils.load(filename)

Load an object.

Parameters

filename (str) – Full path to file where the object is serialized

Returns

obj – Object to be serialized

Return type

object

@authors: Alberto FOSSA’ Giuliana Elena MICELI

Defines useful constants

@authors: Alberto FOSSA’ Giuliana Elena MICELI

class latom.utils.primary.Primary(r, gm)

Bases: object

Primary class defines the characteristic quantities for a given primary body.

Variables

• R (float) – Radius [m]

• GM (float) – Standard gravitational parameter [m^3/s^2]

• g (float) – Surface gravity [m/s]

• tc (float) – Characteristic time [s]

• vc (float) – Characteristic speed [m/s]

• T_circ (float) – Orbital period in a circular orbit with radius equal to R [s]

class latom.utils.primary.Moon

Bases: latom.utils.primary.Primary

Defines the Moon