Interplanetary transfer legs

Interplanetary transfer legs#

class pykep.leg.sims_flanagan(rvs=[[1, 0, 0], [0, 1, 0]], ms=1., throttles=[0, 0, 0, 0, 0, 0], rvf=[[0, 1, 0], [-1, 0, 0]], mf=1., tof=pi / 2, max_thrust=1., veff=1., mu=1., cut=0.5)#

This class represents an interplanetary low-thrust transfer between a starting and a final point in the augmented state-space \([\mathbf r, \mathbf v, m]\). The low-thrust transfer is described by a sequence of equally spaced impulses as described in:

Sims, J., Finlayson, P., Rinderle, E., Vavrina, M. and Kowalkowski, T., 2006, August. Implementation of a low-thrust trajectory optimization algorithm for preliminary design. In AIAA/AAS Astrodynamics specialist conference and exhibit (p. 6746).

The low-thrust transfer will be feasible is the state mismatch equality constraints and the throttle mismatch inequality constraints are satisfied.

Args:

rvs (2D array-like): Cartesian components of the initial position vector and velocity [[xs, ys, zs], [vxs, vys, vzs]]. Defaults to [[1,0,0], [0,1,0]].

ms (float): initial mass. Defaults to 1.

throttles (1D array-like): the Cartesan components of the throttle history [ux1, uy1, uz1, ux2, uy2, uz2, …..]. Defaults to a ballistic, two segments profile [0,0,0,0,0,0].

rvf (2D array-like): Cartesian components of the final position vector and velocity [[xf, yf, zf], [vxf, vyf, vzf]]. Defaults to [[0,1,0], [-1,0,0]].

mf (float): final mass. Defaults to 1.

tof (float): time of flight. Defaults to \(\frac{\pi}{2}\).

max_thrust (float): maximum level for the spacecraft thrust. Defaults to 1.

veff (float): effective velocity of the propulsion system. Defaults to 1.

mu (float): gravitational parameter. Defaults to 1.

cut (float): the leg cut, in [0,1]. It determines the number of forward and backward segments. Defaults to 0.5.

Note

Units need to be consistent.

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> sf = pk.leg.sims_flanagan()

compute_mc_grad()#

Computes the gradients of the mismatch constraints. Indicating the initial augmented state with \(\mathbf x_s = [\mathbf r_s, \mathbf v_s, m_s]\), the final augmented state with \(\mathbf x_f = [\mathbf r_f, \mathbf v_f, m_f]\), the total time of flight with \(T\) and introducing the throttle vector \(\mathbf u = [u_{x0}, u_{y0}, u_{z0}, u_{x1}, u_{y1}, u_{z1} ]\) and \(\mathbf {\tilde u} = [\mathbf u, T]\) (note the time of flight at the end), this method computes the following gradients:

\[\frac{\partial \mathbf {mc}}{\partial \mathbf x_s} \rightarrow (7\times7)\]

\[\frac{\partial \mathbf {mc}}{\partial \mathbf x_f} \rightarrow (7\times7)\]

\[\frac{\partial \mathbf {mc}}{\partial \mathbf {\tilde u}} \rightarrow (7\times(3\mathbf{nseg} + 1))\]

Returns:

tuple [numpy.ndarray, numpy.ndarray, numpy.ndarray]: The three gradients. sizes will be (7,7), (7,7) and (7, 3nseg + 1)

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> sf = pk.leg.sims_flanagan()
>>  sf.throttles = [0.8]*3
>>> sf.compute_mc_grad()

compute_mismatch_constraints()#

In the Sims-Flanagan trajectory leg model, a forward propagation is performed from the starting state as well as a backward from the final state. The state values thus computed need to match in some middle control point. This is typically imposed as 7 independent constraints called mismatch-constraints computed by this method.

Returns:

list [float]: The seven mismatch constraints in the same units used to construct the leg.

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> sf = pk.leg.sims_flanagan()
>>> sf.compute_mismatch_constraints()

compute_tc_grad()#

Computes the gradients of the throttles constraints. Indicating the total time of flight with \(T\) and introducing the throttle vector \(\mathbf u = [u_{x0}, u_{y0}, u_{z0}, u_{x1}, u_{y1}, u_{z1} ]\), this method computes the following gradient:

\[\frac{\partial \mathbf {tc}}{\partial \mathbf u} \rightarrow (\mathbf{nseg} \times3\mathbf{nseg})\]

Returns:

tuple [numpy.ndarray]: The gradient. Size will be (nseg,nseg*3).

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> sf = pk.leg.sims_flanagan()
>>  sf.throttles = [0.8]*3
>>> sf.compute_tc_grad()

compute_throttle_constraints()#

In the Sims-Flanagan trajectory leg model implemented in pykep, we introduce the concept of throttles. Each throttle is defined by three numbers \([u_x, u_y, u_z] \in [0,1]\) indicating that a certain component of the thrust vector has reached a fraction of its maximum allowed value. As a consequence, along the segment along which the throttle is applied, the constraint \(u_x ^2 + u_y ^2 + u_z^2 = 1\), called a throttle constraint, has to be met.

Returns:

list [float]: The throttle constraints.

Examples:

>>> import pykep as pk
>>> import numpy as np
>>> sf = pk.leg.sims_flanagan()
>>  sf.throttles = [0.8]*3
>>> sf.compute_throttle_constraints()

property cut#: The leg cut: it determines the number of forward and backward segments.

property max_thrust#: Maximum spacecraft thruet.

property mf#: Final mass.

property ms#: Initial mass.

property mu#: Central body gravitational parameter.

property nseg#: The total number of segments

property nseg_bck#: The total number of backward segments

property nseg_fwd#: The total number of forward segments

property rvf#: The final position vector and velocity: [[xs, ys, zs], [vxs, vys, vzs]].

property rvs#: The initial position vector and velocity: [[xs, ys, zs], [vxs, vys, vzs]].

property throttles#: The Cartesan components of the throttle history [ux1, uy1, uz1, ux2, uy2, uz2, …..].

property tof#: Time of flight.

property veff#: Effective velocity of the propulsion system (Isp*G0 in the V units of the dynamics)

class pykep.leg.zoh(state0, controls, state1, tgrid, cut, tas)#

This class implements an interplanetary low-thrust transfer between a starting and final state in the augmented state-space \([\mathbf{r}, \mathbf{v}, m]\). The transfer is modelled as a sequence of non-uniform segments along which a continuous and constant (zero-order hold) control acts. The time intervals defining these segments are also provided in tgrid.

The formulation generalises pykep.leg.sims_flanagan to arbitrary dynamics and non-uniform time grids. The dynamics are assumed to be zero-order hold and must be provided as compatible Taylor-adaptive integrators (tas).

Note

The requirements on the tas passed are: a) the first four heyoka parameters must be \(T, i_x, i_y, i_z\), b) the system dimension must be 7.

A transfer is feasible when the state mismatch equality constraints are satisfied. In the intended usage, throttle equality constraints are also enforced to ensure a proper thrust representation as \(T \hat{\mathbf{i}}\) with \(|\hat{\mathbf{i}}| = 1\).

\[i_x^2 + i_y^2 + i_z^2 = 1, \quad \forall \text{segments}\]

Note

The variational integrator ta_var can be None or must have state dimension 84 (7x7 STM + 7x4 control sensitivity) and with the same dynamics as the nominal integrator ta. It is the user that must ensure the suitability of the integrators.

Args:

state0 (array-like): Initial state \([\mathbf{r}_0, \mathbf{v}_0, m_0]\) (length 7)

controls (array-like): Control parameters \([T, i_x, i_y, i_z] \times n_\text{seg}\)

state1 (array-like): Final state \([\mathbf{r}_1, \mathbf{v}_1, m_1]\) (length 7)

tgrid (array-like): Non-uniform time grid (length nseg+1)

cut (float): Forward/backward segment split ratio (0 ≤ cut ≤ 1)

tas (tuple): (ta, ta_var) Taylor-adaptive integrators

ta: Nominal dynamics (state dim 7, pars ≥ 4)

ta_var: Variational dynamics (state dim 84, same pars)

Raises:

ValueError: If state/parameter dimensions mismatch or input lengths are incompatible.

Examples:

import numpy as np
import heyoka as hy
# Define states, controls, time grid
state0 = np.array([1.0, 0.0, 0.0, 0.0, 1.0, 0.0, 1.0])
state1 = np.array([1.2, 0.1, 0.0, 0.0, 0.9, 0.1, 0.95])
controls = np.array([0.022, 0.7, 0.7, 0.1, 0.025, -0.3, 0.8, 0.4])
tgrid = np.array([0.0, 0.5, 1.0, 1.23])
# Get integrators
ta = pk.ta.get_zoh_eq_dyn(tol=1e-16)
ta_var = pk.ta.get_zoh_eq_var(tol=1e-16)
# Construct leg (50/50 split)
leg = zoh(state0, controls, state1, tgrid, cut=0.5, tas=(ta, ta_var))

compute_mc_grad()#

Computes the gradients of the mismatch constraints. Indicating the initial augmented state with \(\mathbf x_s = [\mathbf r_s, \mathbf v_s, m_s]\), the final augmented state with \(\mathbf x_f = [\mathbf r_f, \mathbf v_f, m_f]\), the time grid as \(T_{grid}\) and the introducing the control vector \(\mathbf u = [T_0, i_{x0}, i_{y0}, i_{z0}, T_1, i_{x1}, i_{y1}, i_{z1}]\) (note the time of flight at the end), this method computes the following gradients:

\[\frac{\partial \mathbf {mc}}{\partial \mathbf x_s} \rightarrow (7\times7)\]

\[\frac{\partial \mathbf {mc}}{\partial \mathbf x_f} \rightarrow (7\times7)\]

\[\frac{\partial \mathbf {mc}}{\partial \mathbf u} \rightarrow (7\times(4\mathbf{nseg}))\]

\[\frac{\partial \mathbf {mc}}{\partial \mathbf T_{grid}} \rightarrow (7\times(\mathbf{nseg} + 1))\]

Returns:: tuple [numpy.ndarray, numpy.ndarray, numpy.ndarray, numpy.ndarray]: The four gradients. sizes will be (7,7), (7,7) (7,4nseg) and (7,nseg+1)

compute_tc_grad()#

Computes the gradients of the throttles constraints. Introducing the control vector as \(\mathbf u = [T_0, i_{x0}, i_{y0}, i_{z0}, T_1, i_{x1}, i_{y1}, i_{z1}, ...]\), this method computes the following gradient:

\[\frac{\partial \mathbf {tc}}{\partial \mathbf u} \rightarrow (\mathbf{nseg} \times 4\mathbf{nseg}) \]

Returns:: tuple [numpy.ndarray]: The gradient. Size will be (nseg,4nseg).

get_state_info(N=2)#

This method returns state histories sampled along each ZOH segment, for both the forward and backward propagation parts of the leg. The sampling is performed by calling heyoka.taylor_adaptive.propagate_grid() on a uniformly-spaced grid of N points within each segment.

Args:

N (int): Number of sampling points per segment (including the segment endpoints). The default (N=2) returns only the segment endpoints.

Returns:

tuple: (state_fwd, state_bck) where:

state_fwd (list): List of length nseg_fwd. Each entry contains the sampled 7D state history over the corresponding forward segment (from tgrid[i] to tgrid[i+1]).

state_bck (list): List of length nseg_bck. Each entry contains the sampled 7D state history over the corresponding backward segment (from tgrid[-1-i] to tgrid[-2-i]).

Note

The backward propagation is carried out by integrating from the final time toward earlier times; depending on the backend conventions, the returned grids may thus be time-reversed with respect to a forward-time plot.

Note

This method uses the nominal integrator self.ta and overwrites its internal time, state and the first four parameters (T, i_x, i_y, i_z). If the integrator state must be preserved, call this method on a dedicated copy.

Examples:

ax = pk.plot.make_3Daxis()
fwd, bck = leg.get_state_info(N=100)
for segment in fwd:
    ax.scatter(segment[0,0], segment[0,1], segment[0,2], c='blue')
    ax.plot(segment[:,0], segment[:,1], segment[:,2], c='blue')
for segment in bck:
    ax.scatter(segment[0,0], segment[0,1], segment[0,2], c='darkorange')
    ax.plot(segment[:,0], segment[:,1], segment[:,2], c='darkorange')
ax.view_init(90, -90)

Interplanetary transfer legs

Contents

Interplanetary transfer legs#