latom.guess.guess_2d

@authors: Alberto FOSSA’ Giuliana Elena MICELI

Functions

newton(func, x0[, fprime, args, tol, …])

Find a zero of a real or complex function using the Newton-Raphson (or secant or Halley’s) method.

odeint(func, y0, t[, args, Dfun, col_deriv, …])

Integrate a system of ordinary differential equations.

solve_ivp(fun, t_span, y0[, method, t_eval, …])

Solve an initial value problem for a system of ODEs.

Classes

HohmannTransfer(gm, dep, arr)

HohmannTransfer class implements a two-dimensional Hohmann transfer between two coplanar, circular orbits in the Keplerian two-body approximation.

KepOrb(a, e, i, raan, w, ta, gm)

KepOrb defines a Keplerian Orbit.

PowConstRadius(gm, r0, v0, vf, m0, thrust, isp)

PowConstRadius class implements a two-dimensional powered phase at constant radius from the central attractive body.

TwoDimAscGuess(gm, r, alt, sc)

TwoDimAscGuess provides an initial guess for a two-dimensional ascent trajectory from the Moon surface to a circular LLO.

TwoDimDescGuess(gm, r, alt, sc)

TwoDimDescGuess provides an initial guess for a two-dimensional descent trajectory from a circular LLO to the Moon surface.

TwoDimGuess(gm, r, dep, arr, sc)

TwoDimGuess provides an initial guess for a two-dimensional transfer trajectory combining powered phases at constant radius with Hohmann transfer arcs.

TwoDimLLOGuess(gm, r, dep, arr, sc)

TwoDimLLOGuess provides an initial guess for a two-dimensional ascent or descent trajectory from the Moon surface to a circular Low Lunar Orbit.

TwoDimOrb(gm, **kwargs)

Defines a two-dimensional orbit from its keplerian parameters.
class latom.guess.guess_2d.HohmannTransfer(gm, dep, arr)[source]

Bases: object

HohmannTransfer class implements a two-dimensional Hohmann transfer between two coplanar, circular orbits in the Keplerian two-body approximation.

Parameters

  • gm (float) – Standard gravitational parameter of the central attracting body [m^3/s^2]

  • dep (TwoDimOrb) – TwoDimOrb class instance representing the departure orbit

  • arr (TwoDimOrb) – TwoDimOrb class instance representing the arrival orbit

Variables

  • GM (float) – Standard gravitational parameter of the central attracting body [m^3/s^2]

  • dep (TwoDimOrb) – TwoDimOrb class instance representing the departure orbit

  • arr (TwoDimOrb) – TwoDimOrb class instance representing the arrival orbit

  • ra (float) – Transfer orbit apoapsis radius [m]

  • rp (float) – Transfer orbit periapsis radius [m]

  • transfer (TwoDimOrb) – TwoDimOrb class instance representing the transfer orbit

  • tof (float) – Hohmann transfer time of flight corresponding to half of the transfer period [s]

  • dvp (float) – Impulsive delta-V at periapsis [m/s]

  • dva (float) – Impulsive delta-V at apoapsis [m/s]

  • r (ndarray) – Radius time series for the transfer arc [m]

  • theta (ndarray) – Angle time series for the transfer arc [rad]

  • u (ndarray) – Radial velocity time series for the transfer arc [m/s]

  • v (ndarray) – Tangential velocity time series for the transfer arc [m/s]

  • states (ndarray) – States variables time series for the transfer arc as [r, theta, u, v, m]

  • controls (ndarray) – Control variables time series for the transfer arc as [thrust, alpha]

compute_trajectory(t, t0=0.0, theta0=0.0, m=1.0)[source]

Computes the states and control time series along the transfer trajectory.

Parameters

  • t (ndarray) – Time vector [s]

  • t0 (float) – Initial time [s]

  • theta0 (float, optional) – Initial angle. Default is 0.0 [rad]

  • m (float) – Spacecraft mass. Default is 1.0 [kg]

Returns

root – Solution of the Kepler time of flight equation for the eccentric anomaly in function of the true anomaly

Return type

ndarray

class latom.guess.guess_2d.PowConstRadius(gm, r0, v0, vf, m0, thrust, isp, theta0=0.0, t0=0.0)[source]

Bases: object

PowConstRadius class implements a two-dimensional powered phase at constant radius from the central attractive body.

The trajectory is modeled in the restricted two-body problem framework and the spacecraft engines are supposed to deliver a constant thrust magnitude across the whole phase. Since the radius r is constant, the radial velocity u is always null.

Parameters

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

  • r0 (float) – Radius [m]

  • v0 (float) – Initial tangential velocity [m/s]

  • vf (float) – Final tangential velocity [m/s]

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

  • thrust (float) – Thrust magnitude [N]

  • isp (float) – Specific impulse [s]

  • theta0 (float) – Initial angle [rad]

  • t0 (float) – Initial time [rad]

Variables

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

  • R (float) – Radius [m]

  • v0 (float) – Initial tangential velocity [m/s]

  • vf (float) – Final tangential velocity [m/s]

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

  • T (float) – Thrust magnitude [N]

  • Isp (float) – Specific impulse [s]

  • theta0 (float) – Initial angle [rad]

  • t0 (float) – Initial time [rad]

  • dv_inf (float) – Delta-V required to accelerate the spacecraft from v0 to vf assuming an impulsive burn [m/s]

  • tf (float) – Final time [s]

  • thetaf (float) – Final angle [rad]

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

  • dv (float) – Delta-V required to accelerate the spacecraft from v0 to vf taking into account a finite thrust magnitude equal to T [m/s]

  • t (ndarray) – Time vector [s]

  • 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]

  • m (ndarray) – Spacecraft mass time series [kg]

  • alpha (ndarray) – Thrust direction time series [rad]

  • states (ndarray) – States variables time series as [r, theta, u, v, m]

  • controls (ndarray) – Controls variables time series as [thrust, alpha]

compute_mass(t)[source]

Compute the spacecraft mass time series.

Parameters

t (ndarray) – Time vector [s]

Returns

m – Spacecraft mass time series [kg]

Return type

ndarray

compute_final_time_states()[source]

Compute the required time of flight and the final spacecraft states.

Returns

  • sol_time (tuple) – Solution of the Initial Value Problem (IVP) for the final time. See scipy.integrate.solve_ivp for more details

  • sol_states (tuple) – Solution of the Initial Value Problem (IVP) for the final angle and speed. See scipy.integrate.solve_ivp for more details

compute_trajectory(t_eval)[source]

Compute the spacecraft states and controls variables time series across the powered phase.

Parameters

t_eval (ndarray) – Time vector in which the states and controls are provided [s]

Returns

sol – Solution of the Initial Value Problem (IVP) for the final time. See scipy.integrate.solve_ivp or scipy.integrate.odeint for more details

Return type

tuple or dict

dt_dv(v, t, gm, r, m0, t0, thrust, isp)[source]

ODE for the first derivative of the time t wrt the tangential velocity v.

Parameters

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

  • t (float or ndarray) – Time vector [s]

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

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

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

  • t0 (float) – Initial time [s]

  • thrust (float) – Thrust magnitude [N]

  • isp (float) – Specific impulse [s]

Returns

dt_dv – First derivative of t wrt v [s^2/m]

Return type

float or ndarray

dv_dt(t, v, gm, r, m0, t0, thrust, isp)[source]

ODE for first time derivative of the tangential velocity v.

Parameters

  • t (float or ndarray) – Time vector [s]

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

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

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

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

  • t0 (float) – Initial time [s]

  • thrust (float) – Thrust magnitude [N]

  • isp (float) – Specific impulse [s]

Returns

dv_dt – First time derivative of v [m/s^2]

Return type

float or ndarray

dx_dt(t, x, gm, r, m0, t0, thrust, isp)[source]

Systems of ODEs for the first time derivatives of the angle theta and the tangential velocity v.

Parameters

  • t (float or ndarray) – Time vector [s]

  • x (ndarray) – Angle and tangential velocity time series as [theta, v] [rad, m/s]

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

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

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

  • t0 (float) – Initial time [s]

  • thrust (float) – Thrust magnitude [N]

  • isp (float) – Specific impulse [s]

Returns

x_dot – First time derivatives of theta and v [rad/s, m/s^2]

Return type

ndarray

class latom.guess.guess_2d.TwoDimGuess(gm, r, dep, arr, sc)[source]

Bases: object

TwoDimGuess provides an initial guess for a two-dimensional transfer trajectory combining powered phases at constant radius with Hohmann transfer arcs.

Parameters

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

  • r (float) – Central body equatorial radius [m]

  • dep (TwoDimOrb) – Departure orbit object

  • arr (TwoDimOrb) – Arrival orbit object

  • sc (Spacecraft) – Spacecraft object

Variables

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

  • R (float) – Central body equatorial radius [m]

  • dep (TwoDimOrb) – Departure orbit object

  • ht (HohmannTransfer) – HohmannTransfer object

  • t (ndarray) – Time vector [s]

  • states (ndarray) – States time series as [r, theta, u, v, m]

  • controls (ndarray) – Controls time series as [thrust, alpha]

class latom.guess.guess_2d.TwoDimLLOGuess(gm, r, dep, arr, sc)[source]

Bases: latom.guess.guess_2d.TwoDimGuess

TwoDimLLOGuess provides an initial guess for a two-dimensional ascent or descent trajectory from the Moon surface to a circular Low Lunar Orbit.

The approximate transfer consists into two powered phases at constant radius and an Hohmann transfer.

Parameters

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

  • r (float) – Central body equatorial radius [m]

  • dep (TwoDimOrb) – Departure orbit object

  • arr (TwoDimOrb) – Arrival orbit object

  • sc (Spacecraft) – Spacecraft object

Variables
compute_trajectory(fix_final=False, **kwargs)[source]

Computes the states and controls time series for a given time vector or number of equally space nodes.

Parameters

fix_final (bool, optional) – True if the final angle is fixed, False otherwise. Default is False

Other Parameters

  • t_eval (ndarray) – Time vector in which states and controls are computed [s]

  • nb_nodes (int) – Number of equally space nodes in time in which states and controls are computed

class latom.guess.guess_2d.TwoDimAscGuess(gm, r, alt, sc)[source]

Bases: latom.guess.guess_2d.TwoDimLLOGuess

TwoDimAscGuess provides an initial guess for a two-dimensional ascent trajectory from the Moon surface to a circular LLO.

The approximate transfer comprises a first powered phase at constant radius equal to the Moon one, an Hohmann transfer and a second powered phase at constant radius equal to the LLO one.

Parameters

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

  • r (float) – Central body equatorial radius [m]

  • alt (float) – LLO altitude [m]

  • sc (Spacecraft) – Spacecraft object

Variables

  • pow1 (PowConstRadius) – First powered phase at constant radius

  • pow2 (PowConstRadius) – Second powered phase at constant radius

  • tf (float) – Final time [s]

compute_trajectory(fix_final=False, **kwargs)

Computes the states and controls time series for a given time vector or number of equally space nodes.

Parameters

fix_final (bool, optional) – True if the final angle is fixed, False otherwise. Default is False

Other Parameters

  • t_eval (ndarray) – Time vector in which states and controls are computed [s]

  • nb_nodes (int) – Number of equally space nodes in time in which states and controls are computed

class latom.guess.guess_2d.TwoDimDescGuess(gm, r, alt, sc)[source]

Bases: latom.guess.guess_2d.TwoDimLLOGuess

TwoDimDescGuess provides an initial guess for a two-dimensional descent trajectory from a circular LLO to the Moon surface.

The approximate transfer comprises a first powered phase at constant radius equal to the LLO one, an Hohmann transfer and a second powered phase at constant radius equal to the Moon one.

Parameters

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

  • r (float) – Central body equatorial radius [m]

  • alt (float) – LLO altitude [m]

  • sc (Spacecraft) – Spacecraft object

Variables

  • pow1 (PowConstRadius) – First powered phase at constant radius

  • pow2 (PowConstRadius) – Second powered phase at constant radius

  • tf (float) – Final time [s]

compute_trajectory(fix_final=False, **kwargs)

Computes the states and controls time series for a given time vector or number of equally space nodes.

Parameters

fix_final (bool, optional) – True if the final angle is fixed, False otherwise. Default is False

Other Parameters

  • t_eval (ndarray) – Time vector in which states and controls are computed [s]

  • nb_nodes (int) – Number of equally space nodes in time in which states and controls are computed