def point_taylor(r, v, m, u, t, mu, veff, N=60, units=1):
from PyKEP import propagate_taylor
# We define the integration time ...
dt = t / (N - 1)
mm = []
# ... and calcuate the cartesian components for r
x = [0.0] * N
y = [0.0] * N
z = [0.0] * N
# We calculate the spacecraft position at each dt
for i in range(N):
x[i] = r[0] / units
y[i] = r[1] / units
z[i] = r[2] / units
mm = mm + [m]
r, v, m = propagate_taylor(r, v, m, u, dt, mu, veff, -10, -10)
tt = range(N)
tt = [e*dt/(60*60*24) for e in tt]
return x,y,z,tt,mm
def plot_taylor(r, v, m, u, t, mu, veff, N=60, units=1, color='b', legend=False, ax=None):
"""
ax = plot_taylor(r, v, m, u, t, mu, veff, N=60, units=1, color='b', legend=False, ax=None):
- ax: 3D axis object created using fig.gca(projection='3d')
- r: initial position (cartesian coordinates)
- v: initial velocity (cartesian coordinates)
- m: initial mass
- u: cartesian components for the constant thrust
- t: propagation time
- mu: gravitational parameter
- veff: the product Isp * g0
- N: number of points to be plotted along one arc
- units: the length unit to be used in the plot
- color: matplotlib color to use to plot the line
- legend: when True it plots also the legend
Plots the result of a taylor propagation of constant thrust
"""
from PyKEP import propagate_taylor
import matplotlib.pyplot as plt
if ax is None:
fig = plt.figure()
axis = fig.gca(projection='3d')
else:
axis = ax
# We define the integration time ...
dt = t / (N - 1)
# ... and calcuate the cartesian components for r
x = [0.0] * N
y = [0.0] * N
z = [0.0] * N
# We calculate the spacecraft position at each dt
for i in range(N):
x[i] = r[0] / units
y[i] = r[1] / units
z[i] = r[2] / units
r, v, m = propagate_taylor(r, v, m, u, dt, mu, veff, -10, -10)
# And we plot
if legend:
label = 'constant thrust arc'
else:
label = None
axis.plot(x, y, z, c=color, label=label)
if legend:
axis.legend()
if ax is None: # show only if axis is not set
plt.show()
return axis
def plot_taylor(ax,r,v,m,u,t,mu, veff, N=60, units = 1, color = 'b', legend = False): """ Plots the result of a taylor propagation of constant thrust USAGE: plot_taylor(ax,r,v,m,u,t,mu, veff, N=60, units = 1, color = 'b', legend = False): * ax: 3D axis object created using fig.gca(projection='3d') * r: initial position (cartesian coordinates) * v: initial velocity (cartesian coordinates) * m: initial mass * u: cartesian components for the constant thrust * t: propagation time * mu: gravitational parameter * veff: the product Isp * g0 * N: number of points to be plotted along one arc * units: the length unit to be used in the plot * color: matplotlib color to use to plot the line * legend: when True it plots also the legend """ from PyKEP import propagate_taylor #We define the integration time ... dt = t / (N-1) #... and calcuate the cartesian components for r x = [0.0]*N y = [0.0]*N z = [0.0]*N #We calculate the spacecraft position at each dt for i in range(N): x[i] = r[0]/units y[i] = r[1]/units z[i] = r[2]/units r,v,m = propagate_taylor(r,v,m,u,dt,mu,veff,-10,-10) #And we plot if legend: label = 'constant thrust arc' else: label = None ax.plot(x, y, z, c=color, label=label) if legend: ax.legend()
def plot_sf_leg(leg,
N=5,
units=1,
color='b',
legend=False,
plot_line=True,
ax=None):
"""
ax = plot_sf_leg(leg, N=5, units=1, color='b', legend=False, no_trajectory=False, ax=None):
- ax: 3D axis object created using fig.gca(projection='3d')
- leg: a PyKEP.sims_flanagan.leg
- N: number of points to be plotted along one arc
- units: the length unit to be used in the plot
- color: matplotlib color to use to plot the trajectory and the grid points
- legend when True it plots also the legend
- plot_line: when True plots also the trajectory (between mid-points and grid points)
Plots a Sims-Flanagan leg
Example::
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.gca(projection='3d')
t1 = epoch(0)
pl = planet_ss('earth')
rE,vE = pl.eph(t1)
plot_planet(pl,t0=t1, units=AU, ax=ax)
t2 = epoch(440)
pl = planet_ss('mars')
rM, vM = pl.eph(t2)
plot_planet(pl,t0=t2, units=AU, ax=ax)
sc = sims_flanagan.spacecraft(4500,0.5,2500)
x0 = sims_flanagan.sc_state(rE,vE,sc.mass)
xe = sims_flanagan.sc_state(rM,vM,sc.mass)
l = sims_flanagan.leg(t1,x0,[1,0,0]*5,t2,xe,sc,MU_SUN)
plot_sf_leg(l, units=AU, ax=ax)
"""
from PyKEP import propagate_lagrangian, AU, DAY2SEC, G0, propagate_taylor
import numpy as np
from scipy.linalg import norm
from math import exp
import matplotlib.pylab as plt
from mpl_toolkits.mplot3d import Axes3D
if ax is None:
fig = plt.figure()
axis = fig.gca(projection='3d')
else:
axis = ax
# We compute the number of segments for forward and backward propagation
n_seg = len(leg.get_throttles())
fwd_seg = (n_seg + 1) // 2
back_seg = n_seg // 2
# We extract information on the spacecraft
sc = leg.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
# And on the leg
throttles = leg.get_throttles()
mu = leg.get_mu()
# Forward propagation
# x,y,z contain the cartesian components of all points (grid+midpints)
x = [0.0] * (fwd_seg * 2 + 1)
y = [0.0] * (fwd_seg * 2 + 1)
z = [0.0] * (fwd_seg * 2 + 1)
state = leg.get_xi()
# Initial conditions
r = state.r
v = state.v
m = state.m
x[0] = r[0] / units
y[0] = r[1] / units
z[0] = r[2] / units
# We compute all points by propagation
for i, t in enumerate(throttles[:fwd_seg]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
# Keplerian propagation and dV application
if leg.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
if plot_line:
plot_kepler(r,
v,
dt / 2,
mu,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v = propagate_lagrangian(r, v, dt / 2, mu)
x[2 * i + 1] = r[0] / units
y[2 * i + 1] = r[1] / units
z[2 * i + 1] = r[2] / units
# v= v+dV
v = [a + b for a, b in zip(v, dV)]
if plot_line:
plot_kepler(r,
v,
dt / 2,
mu,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v = propagate_lagrangian(r, v, dt / 2, mu)
x[2 * i + 2] = r[0] / units
y[2 * i + 2] = r[1] / units
z[2 * i + 2] = r[2] / units
m *= exp(-norm(dV) / isp / G0)
# Taylor propagation of constant thrust u
else:
u = [max_thrust * dumb for dumb in t.value]
if plot_line:
plot_taylor(r,
v,
m,
u,
dt / 2,
mu,
isp * G0,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v, m = propagate_taylor(r, v, m, u, dt / 2, mu, isp * G0, -10,
-10)
x[2 * i + 1] = r[0] / units
y[2 * i + 1] = r[1] / units
z[2 * i + 1] = r[2] / units
if plot_line:
plot_taylor(r,
v,
m,
u,
dt / 2,
mu,
isp * G0,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v, m = propagate_taylor(r, v, m, u, dt / 2, mu, isp * G0, -10,
-10)
x[2 * i + 2] = r[0] / units
y[2 * i + 2] = r[1] / units
z[2 * i + 2] = r[2] / units
x_grid = x[::2]
y_grid = y[::2]
z_grid = z[::2]
x_midpoint = x[1::2]
y_midpoint = y[1::2]
z_midpoint = z[1::2]
axis.scatter(x_grid[:-1],
y_grid[:-1],
z_grid[:-1],
label='nodes',
marker='o')
axis.scatter(x_midpoint,
y_midpoint,
z_midpoint,
label='mid-points',
marker='x')
axis.scatter(x_grid[-1],
y_grid[-1],
z_grid[-1],
marker='^',
c='y',
label='mismatch point')
# Backward propagation
# x,y,z will contain the cartesian components of
x = [0.0] * (back_seg * 2 + 1)
y = [0.0] * (back_seg * 2 + 1)
z = [0.0] * (back_seg * 2 + 1)
state = leg.get_xf()
# Final conditions
r = state.r
v = state.v
m = state.m
x[-1] = r[0] / units
y[-1] = r[1] / units
z[-1] = r[2] / units
for i, t in enumerate(throttles[-1:-back_seg - 1:-1]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
if leg.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
if plot_line:
plot_kepler(r,
v,
-dt / 2,
mu,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
x[-2 * i - 2] = r[0] / units
y[-2 * i - 2] = r[1] / units
z[-2 * i - 2] = r[2] / units
# v= v+dV
v = [a - b for a, b in zip(v, dV)]
if plot_line:
plot_kepler(r,
v,
-dt / 2,
mu,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
x[-2 * i - 3] = r[0] / units
y[-2 * i - 3] = r[1] / units
z[-2 * i - 3] = r[2] / units
m *= exp(norm(dV) / isp / G0)
else:
u = [max_thrust * dumb for dumb in t.value]
if plot_line:
plot_taylor(r,
v,
m,
u,
-dt / 2,
mu,
isp * G0,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v, m = propagate_taylor(r, v, m, u, -dt / 2, mu, isp * G0, -10,
-10)
x[-2 * i - 2] = r[0] / units
y[-2 * i - 2] = r[1] / units
z[-2 * i - 2] = r[2] / units
if plot_line:
plot_taylor(r,
v,
m,
u,
-dt / 2,
mu,
isp * G0,
N=N,
units=units,
color=(alpha, 0, 1 - alpha),
ax=axis)
r, v, m = propagate_taylor(r, v, m, u, -dt / 2, mu, isp * G0, -10,
-10)
x[-2 * i - 3] = r[0] / units
y[-2 * i - 3] = r[1] / units
z[-2 * i - 3] = r[2] / units
x_grid = x[::2]
y_grid = y[::2]
z_grid = z[::2]
x_midpoint = x[1::2]
y_midpoint = y[1::2]
z_midpoint = z[1::2]
axis.scatter(x_grid[1:], y_grid[1:], z_grid[1:], marker='o')
axis.scatter(x_midpoint, y_midpoint, z_midpoint, marker='x')
axis.scatter(x_grid[0], y_grid[0], z_grid[0], marker='^', c='y')
if legend:
axis.legend()
if ax is None: # show only if axis is not set
plt.show()
return axis
def plot_taylor(r,
v,
m,
u,
t,
mu,
veff,
N=60,
units=1,
color='b',
legend=False,
ax=None):
"""
ax = plot_taylor(r, v, m, u, t, mu, veff, N=60, units=1, color='b', legend=False, ax=None):
- ax: 3D axis object created using fig.gca(projection='3d')
- r: initial position (cartesian coordinates)
- v: initial velocity (cartesian coordinates)
- m: initial mass
- u: cartesian components for the constant thrust
- t: propagation time
- mu: gravitational parameter
- veff: the product Isp * g0
- N: number of points to be plotted along one arc
- units: the length unit to be used in the plot
- color: matplotlib color to use to plot the line
- legend: when True it plots also the legend
Plots the result of a taylor propagation of constant thrust
"""
from PyKEP import propagate_taylor
import matplotlib.pyplot as plt
if ax is None:
fig = plt.figure()
axis = fig.gca(projection='3d')
else:
axis = ax
# We define the integration time ...
dt = t / (N - 1)
# ... and calcuate the cartesian components for r
x = [0.0] * N
y = [0.0] * N
z = [0.0] * N
# We calculate the spacecraft position at each dt
for i in range(N):
x[i] = r[0] / units
y[i] = r[1] / units
z[i] = r[2] / units
r, v, m = propagate_taylor(r, v, m, u, dt, mu, veff, -10, -10)
# And we plot
if legend:
label = 'constant thrust arc'
else:
label = None
axis.plot(x, y, z, c=color, label=label)
if legend:
axis.legend()
if ax is None: # show only if axis is not set
plt.show()
return axis
def plot_sf_leg(leg, N=5, units=1, color='b', legend=False, plot_line=True, plot_segments=True, ax=None):
"""
ax = plot_sf_leg(leg, N=5, units=1, color='b', legend=False, no_trajectory=False, ax=None):
- ax: 3D axis object created using fig.gca(projection='3d')
- leg: a PyKEP.sims_flanagan.leg
- N: number of points to be plotted along one arc
- units: the length unit to be used in the plot
- color: matplotlib color to use to plot the trajectory and the grid points
- legend when True it plots also the legend
- plot_line: when True plots also the trajectory (between mid-points and grid points)
Plots a Sims-Flanagan leg
Example::
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.gca(projection='3d')
t1 = epoch(0)
pl = planet_ss('earth')
rE,vE = pl.eph(t1)
plot_planet(pl,t0=t1, units=AU, ax=ax)
t2 = epoch(440)
pl = planet_ss('mars')
rM, vM = pl.eph(t2)
plot_planet(pl,t0=t2, units=AU, ax=ax)
sc = sims_flanagan.spacecraft(4500,0.5,2500)
x0 = sims_flanagan.sc_state(rE,vE,sc.mass)
xe = sims_flanagan.sc_state(rM,vM,sc.mass)
l = sims_flanagan.leg(t1,x0,[1,0,0]*5,t2,xe,sc,MU_SUN)
plot_sf_leg(l, units=AU, ax=ax)
"""
from PyKEP import propagate_lagrangian, AU, DAY2SEC, G0, propagate_taylor
import numpy as np
from scipy.linalg import norm
from math import exp
import matplotlib.pylab as plt
from mpl_toolkits.mplot3d import Axes3D
if ax is None:
fig = plt.figure()
axis = fig.gca(projection='3d')
else:
axis = ax
# We compute the number of segments for forward and backward propagation
n_seg = len(leg.get_throttles())
fwd_seg = (n_seg + 1) // 2
back_seg = n_seg // 2
# We extract information on the spacecraft
sc = leg.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
# And on the leg
throttles = leg.get_throttles()
mu = leg.get_mu()
# Forward propagation
# x,y,z contain the cartesian components of all points (grid+midpints)
x = [0.0] * (fwd_seg * 2 + 1)
y = [0.0] * (fwd_seg * 2 + 1)
z = [0.0] * (fwd_seg * 2 + 1)
state = leg.get_xi()
# Initial conditions
r = state.r
v = state.v
m = state.m
x[0] = r[0] / units
y[0] = r[1] / units
z[0] = r[2] / units
# We compute all points by propagation
for i, t in enumerate(throttles[:fwd_seg]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
# Keplerian propagation and dV application
if leg.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
if plot_line:
plot_kepler(r, v, dt / 2, mu, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v = propagate_lagrangian(r, v, dt / 2, mu)
x[2 * i + 1] = r[0] / units
y[2 * i + 1] = r[1] / units
z[2 * i + 1] = r[2] / units
# v= v+dV
v = [a + b for a, b in zip(v, dV)]
if plot_line:
plot_kepler(r, v, dt / 2, mu, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v = propagate_lagrangian(r, v, dt / 2, mu)
x[2 * i + 2] = r[0] / units
y[2 * i + 2] = r[1] / units
z[2 * i + 2] = r[2] / units
m *= exp(-norm(dV) / isp / G0)
# Taylor propagation of constant thrust u
else:
u = [max_thrust * dumb for dumb in t.value]
if plot_line:
plot_taylor(r, v, m, u, dt / 2, mu, isp * G0, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v, m = propagate_taylor(r, v, m, u, dt / 2, mu, isp * G0, -12, -12)
x[2 * i + 1] = r[0] / units
y[2 * i + 1] = r[1] / units
z[2 * i + 1] = r[2] / units
if plot_line:
plot_taylor(r, v, m, u, dt / 2, mu, isp * G0, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v, m = propagate_taylor(r, v, m, u, dt / 2, mu, isp * G0, -12, -12)
x[2 * i + 2] = r[0] / units
y[2 * i + 2] = r[1] / units
z[2 * i + 2] = r[2] / units
x_grid = x[::2]
y_grid = y[::2]
z_grid = z[::2]
x_midpoint = x[1::2]
y_midpoint = y[1::2]
z_midpoint = z[1::2]
if plot_segments:
axis.scatter(x_grid[:-1], y_grid[:-1], z_grid[:-1], label='nodes', marker='o')
axis.scatter(x_midpoint, y_midpoint, z_midpoint, label='mid-points', marker='x')
axis.scatter(x_grid[-1], y_grid[-1], z_grid[-1], marker='^', c='y', label='mismatch point')
# Backward propagation
# x,y,z will contain the cartesian components of
x = [0.0] * (back_seg * 2 + 1)
y = [0.0] * (back_seg * 2 + 1)
z = [0.0] * (back_seg * 2 + 1)
state = leg.get_xf()
# Final conditions
r = state.r
v = state.v
m = state.m
x[-1] = r[0] / units
y[-1] = r[1] / units
z[-1] = r[2] / units
for i, t in enumerate(throttles[-1:-back_seg - 1:-1]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
if leg.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
if plot_line:
plot_kepler(r, v, -dt / 2, mu, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
x[-2 * i - 2] = r[0] / units
y[-2 * i - 2] = r[1] / units
z[-2 * i - 2] = r[2] / units
# v= v+dV
v = [a - b for a, b in zip(v, dV)]
if plot_line:
plot_kepler(r, v, -dt / 2, mu, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
x[-2 * i - 3] = r[0] / units
y[-2 * i - 3] = r[1] / units
z[-2 * i - 3] = r[2] / units
m *= exp(norm(dV) / isp / G0)
else:
u = [max_thrust * dumb for dumb in t.value]
if plot_line:
plot_taylor(r, v, m, u, -dt / 2, mu, isp * G0, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v, m = propagate_taylor(r, v, m, u, -dt / 2, mu, isp * G0, -12, -12)
x[-2 * i - 2] = r[0] / units
y[-2 * i - 2] = r[1] / units
z[-2 * i - 2] = r[2] / units
if plot_line:
plot_taylor(r, v, m, u, -dt / 2, mu, isp * G0, N=N, units=units, color=(alpha, 0, 1 - alpha), ax=axis)
r, v, m = propagate_taylor(r, v, m, u, -dt / 2, mu, isp * G0, -12, -12)
x[-2 * i - 3] = r[0] / units
y[-2 * i - 3] = r[1] / units
z[-2 * i - 3] = r[2] / units
x_grid = x[::2]
y_grid = y[::2]
z_grid = z[::2]
x_midpoint = x[1::2]
y_midpoint = y[1::2]
z_midpoint = z[1::2]
if plot_segments:
axis.scatter(x_grid[1:], y_grid[1:], z_grid[1:], marker='o')
axis.scatter(x_midpoint, y_midpoint, z_midpoint, marker='x')
axis.scatter(x_grid[0], y_grid[0], z_grid[0], marker='^', c='y')
if legend:
axis.legend()
if ax is None: # show only if axis is not set
plt.show()
return axis
def _leg_eph(self, t, debug=False):
"""
Computes the ephemerides (r, v) along the leg. Should only be called on high_fidelity legs having
no state mismatch. Otherwise the values returned will not correspond to physical quantities.
- t: epoch (either a pykep epoch, or assumes mjd2000). This value must be between self.get_ti() and self.get_tf()
"""
from PyKEP import epoch, propagate_taylor, G0, DAY2SEC
from bisect import bisect
if isinstance(t, epoch):
t0 = t.mjd2000
else:
t0 = t
# We extract the leg states
t_grid, r, v, m = self.get_states()
# We check that requested epoch is valid
if (t0 < t_grid[0]) or (t0 > t_grid[-1]):
raise ValueError("The requested epoch is out of bounds")
# We check that the leg is high fidelity, otherwise this makes little sense
if not self.high_fidelity:
raise ValueError("The eph method works only for high fidelity legs at the moment")
# Extract some information from the leg
mu = self.get_mu()
sc = self.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
t_grid, r, v, m = self.get_states()
# If by chance its in the grid node, we are done
if t0 in t_grid:
idx = t_grid.index(t0)
if debug:
raise ValueError("DEBUG!!!!")
return r[idx], v[idx], m[idx]
# Find the index to start from (need to account for the midpoint repetition)
idx = bisect(t_grid, t0) - 1
# We compute the number of segments of forward propagation
n_seg = len(self.get_throttles())
fwd_seg = (n_seg + 1) // 2
# We start with the backward propagation cases
if idx > 2 * fwd_seg:
r0 = r[idx + 1]
v0 = v[idx + 1]
m0 = m[idx + 1]
idx_thrust = (idx - 1) / 2
dt_int = (t_grid[idx + 1] - t0) * DAY2SEC
th = self.get_throttles()[idx_thrust].value
if debug:
raise ValueError("DEBUG!!!!")
return propagate_taylor(r0, v0, m0, [d * max_thrust for d in th], -dt_int, mu, isp * G0, -12, -12)
# Find the index to start from
idx = bisect(t_grid, t0) - 1
r0 = r[idx]
v0 = v[idx]
m0 = m[idx]
idx_thrust = idx / 2
dt_int = (t0 - t_grid[idx]) * DAY2SEC
th = self.get_throttles()[idx_thrust].value
if debug:
raise ValueError("DEBUG!!!!")
return propagate_taylor(r0, v0, m0, [d * max_thrust for d in th], dt_int, mu, isp * G0, -12, -12)
def _leg_get_states(self):
"""
Returns the spacecraft states (t,r,v,m) at the leg grid points
Examples::
times,r,v,m = PyKEP.sims_flanagan.leg.get_states()
"""
from PyKEP import propagate_lagrangian, AU, DAY2SEC, G0, propagate_taylor
import numpy as np
from scipy.linalg import norm
from math import exp
# We compute the number of segments for forward and backward propagation
n_seg = len(self.get_throttles())
fwd_seg = (n_seg + 1) // 2
back_seg = n_seg // 2
# We extract information on the spacecraft
sc = self.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
# And on the leg
throttles = self.get_throttles()
mu = self.get_mu()
# time grid
t_grid = [0.0] * (n_seg * 2 + 2)
# Forward propagation
# x,y,z contain the cartesian components of all points (grid+midpints)
x = [0.0] * (fwd_seg * 2 + 1)
y = [0.0] * (fwd_seg * 2 + 1)
z = [0.0] * (fwd_seg * 2 + 1)
vx = [0.0] * (fwd_seg * 2 + 1)
vy = [0.0] * (fwd_seg * 2 + 1)
vz = [0.0] * (fwd_seg * 2 + 1)
mass = [0.0] * (fwd_seg * 2 + 1)
state = self.get_xi()
# Initial conditions
r = state.r
v = state.v
m = state.m
x[0], y[0], z[0] = r
vx[0], vy[0], vz[0] = v
mass[0] = m
# We compute all points by propagation
for i, t in enumerate(throttles[:fwd_seg]):
t_grid[2 * i] = t.start.mjd2000
t_grid[2 * i + 1] = t.start.mjd2000 + \
(t.end.mjd2000 - t.start.mjd2000) / 2.
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
# Keplerian propagation and dV application
if self.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
r, v = propagate_lagrangian(r, v, dt / 2, mu)
x[2 * i + 1], y[2 * i + 1], z[2 * i + 1] = r
vx[2 * i + 1], vy[2 * i + 1], vz[2 * i + 1] = v
mass[2 * i + 1] = m
# v= v+dV
v = [a + b for a, b in zip(v, dV)]
r, v = propagate_lagrangian(r, v, dt / 2, mu)
m *= exp(-norm(dV) / isp / G0)
x[2 * i + 2], y[2 * i + 2], z[2 * i + 2] = r
vx[2 * i + 2], vy[2 * i + 2], vz[2 * i + 2] = v
mass[2 * i + 2] = m
# Taylor propagation of constant thrust u
else:
u = [max_thrust * dumb for dumb in t.value]
r, v, m = propagate_taylor(
r, v, m, u, dt / 2, mu, isp * G0, -12, -12)
x[2 * i + 1], y[2 * i + 1], z[2 * i + 1] = r
vx[2 * i + 1], vy[2 * i + 1], vz[2 * i + 1] = v
mass[2 * i + 1] = m
r, v, m = propagate_taylor(
r, v, m, u, dt / 2, mu, isp * G0, -12, -12)
x[2 * i + 2], y[2 * i + 2], z[2 * i + 2] = r
vx[2 * i + 2], vy[2 * i + 2], vz[2 * i + 2] = v
mass[2 * i + 2] = m
t_grid[2 * i + 2] = t.end.mjd2000
# Backward propagation
# x,y,z will contain the cartesian components of
x_back = [0.123] * (back_seg * 2 + 1)
y_back = [0.123] * (back_seg * 2 + 1)
z_back = [0.123] * (back_seg * 2 + 1)
vx_back = [0.0] * (back_seg * 2 + 1)
vy_back = [0.0] * (back_seg * 2 + 1)
vz_back = [0.0] * (back_seg * 2 + 1)
mass_back = [0.0] * (back_seg * 2 + 1)
state = self.get_xf()
# Final conditions
r = state.r
v = state.v
m = state.m
x_back[-1], y_back[-1], z_back[-1] = r
vx_back[-1], vy_back[-1], vz_back[-1] = v
mass_back[-1] = m
for i, t in enumerate(throttles[-1:-back_seg - 1:-1]):
t_grid[-2 * i - 2] = t.end.mjd2000 - \
(t.end .mjd2000 - t.start.mjd2000) / 2.
t_grid[-2 * i - 1] = t.end.mjd2000
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
if self.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
x_back[-2 * i - 2], y_back[-2 * i - 2], z_back[-2 * i - 2] = r
vx_back[-2 * i - 2], vy_back[-2 * i - 2], vz_back[-2 * i - 2] = v
mass_back[-2 * i - 2] = m
# v= v+dV
v = [a - b for a, b in zip(v, dV)]
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
m *= exp(norm(dV) / isp / G0)
x_back[-2 * i - 3], y_back[-2 * i - 3], z_back[-2 * i - 3] = r
vx_back[-2 * i - 3], vy_back[-2 * i - 3], vz_back[-2 * i - 3] = v
mass_back[-2 * i - 3] = m
else:
u = [max_thrust * dumb for dumb in t.value]
r, v, m = propagate_taylor(
r, v, m, u, -dt / 2, mu, isp * G0, -12, -12)
x_back[-2 * i - 2], y_back[-2 * i - 2], z_back[-2 * i - 2] = r
vx_back[-2 * i - 2], vy_back[-2 * i - 2], vz_back[-2 * i - 2] = v
mass_back[-2 * i - 2] = m
r, v, m = propagate_taylor(
r, v, m, u, -dt / 2, mu, isp * G0, -12, -12)
x_back[-2 * i - 3], y_back[-2 * i - 3], z_back[-2 * i - 3] = r
vx_back[-2 * i - 3], vy_back[-2 * i - 3], vz_back[-2 * i - 3] = v
mass_back[-2 * i - 3] = m
t_grid[-2 * i - 3] = t.start.mjd2000
x = x + x_back
y = y + y_back
z = z + z_back
vx = vx + vx_back
vy = vy + vy_back
vz = vz + vz_back
mass = mass + mass_back
return t_grid, list(zip(x, y, z)), list(zip(vx, vy, vz)), mass
def _get_states(self):
"""
Returns the spacecraft states (t,r,v,m) at the leg grid points
Examples::
times,r,v,m = PyKEP.sims_flanagan.leg.get_states()
"""
from PyKEP import propagate_lagrangian, AU, DAY2SEC, G0, propagate_taylor
import numpy as np
from scipy.linalg import norm
from math import exp
# We compute the number of segments for forward and backward propagation
n_seg = len(self.get_throttles())
fwd_seg = (n_seg + 1) / 2
back_seg = n_seg / 2
# We extract information on the spacecraft
sc = self.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
# And on the leg
throttles = self.get_throttles()
mu = self.get_mu()
# time grid
t_grid = [0.0] * (n_seg * 2 + 2)
# Forward propagation
# x,y,z contain the cartesian components of all points (grid+midpints)
x = [0.0] * (fwd_seg * 2 + 1)
y = [0.0] * (fwd_seg * 2 + 1)
z = [0.0] * (fwd_seg * 2 + 1)
vx = [0.0] * (fwd_seg * 2 + 1)
vy = [0.0] * (fwd_seg * 2 + 1)
vz = [0.0] * (fwd_seg * 2 + 1)
mass = [0.0] * (fwd_seg * 2 + 1)
state = self.get_xi()
# Initial conditions
r = state.r
v = state.v
m = state.m
x[0], y[0], z[0] = r
vx[0], vy[0], vz[0] = v
mass[0] = m
# We compute all points by propagation
for i, t in enumerate(throttles[:fwd_seg]):
t_grid[2 * i] = t.start.mjd2000
t_grid[2 * i + 1] = t.start.mjd2000 + \
(t.end.mjd2000 - t.start.mjd2000) / 2.
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
# Keplerian propagation and dV application
if self.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
r, v = propagate_lagrangian(r, v, dt / 2, mu)
x[2 * i + 1], y[2 * i + 1], z[2 * i + 1] = r
vx[2 * i + 1], vy[2 * i + 1], vz[2 * i + 1] = v
mass[2 * i + 1] = m
# v= v+dV
v = [a + b for a, b in zip(v, dV)]
r, v = propagate_lagrangian(r, v, dt / 2, mu)
m *= exp(-norm(dV) / isp / G0)
x[2 * i + 2], y[2 * i + 2], z[2 * i + 2] = r
vx[2 * i + 2], vy[2 * i + 2], vz[2 * i + 2] = v
mass[2 * i + 2] = m
# Taylor propagation of constant thrust u
else:
u = [max_thrust * dumb for dumb in t.value]
r, v, m = propagate_taylor(r, v, m, u, dt / 2, mu, isp * G0, -10,
-10)
x[2 * i + 1], y[2 * i + 1], z[2 * i + 1] = r
vx[2 * i + 1], vy[2 * i + 1], vz[2 * i + 1] = v
mass[2 * i + 1] = m
r, v, m = propagate_taylor(r, v, m, u, dt / 2, mu, isp * G0, -10,
-10)
x[2 * i + 2], y[2 * i + 2], z[2 * i + 2] = r
vx[2 * i + 2], vy[2 * i + 2], vz[2 * i + 2] = v
mass[2 * i + 2] = m
t_grid[2 * i + 2] = t.end.mjd2000
# Backward propagation
# x,y,z will contain the cartesian components of
x_back = [0.123] * (back_seg * 2 + 1)
y_back = [0.123] * (back_seg * 2 + 1)
z_back = [0.123] * (back_seg * 2 + 1)
vx_back = [0.0] * (back_seg * 2 + 1)
vy_back = [0.0] * (back_seg * 2 + 1)
vz_back = [0.0] * (back_seg * 2 + 1)
mass_back = [0.0] * (back_seg * 2 + 1)
state = self.get_xf()
# Final conditions
r = state.r
v = state.v
m = state.m
x_back[-1], y_back[-1], z_back[-1] = r
vx_back[-1], vy_back[-1], vz_back[-1] = v
mass_back[-1] = m
for i, t in enumerate(throttles[-1:-back_seg - 1:-1]):
t_grid[-2 * i - 2] = t.end.mjd2000 - \
(t.end.mjd2000 - t.start.mjd2000) / 2.
t_grid[-2 * i - 1] = t.end.mjd2000
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
if self.high_fidelity is False:
dV = [max_thrust / m * dt * dumb for dumb in t.value]
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
x_back[-2 * i - 2], y_back[-2 * i - 2], z_back[-2 * i - 2] = r
vx_back[-2 * i - 2], vy_back[-2 * i - 2], vz_back[-2 * i - 2] = v
mass_back[-2 * i - 2] = m
# v= v+dV
v = [a - b for a, b in zip(v, dV)]
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
m *= exp(norm(dV) / isp / G0)
x_back[-2 * i - 3], y_back[-2 * i - 3], z_back[-2 * i - 3] = r
vx_back[-2 * i - 3], vy_back[-2 * i - 3], vz_back[-2 * i - 3] = v
mass_back[-2 * i - 3] = m
else:
u = [max_thrust * dumb for dumb in t.value]
r, v, m = propagate_taylor(r, v, m, u, -dt / 2, mu, isp * G0, -10,
-10)
x_back[-2 * i - 2], y_back[-2 * i - 2], z_back[-2 * i - 2] = r
vx_back[-2 * i - 2], vy_back[-2 * i - 2], vz_back[-2 * i - 2] = v
mass_back[-2 * i - 2] = m
r, v, m = propagate_taylor(r, v, m, u, -dt / 2, mu, isp * G0, -10,
-10)
x_back[-2 * i - 3], y_back[-2 * i - 3], z_back[-2 * i - 3] = r
vx_back[-2 * i - 3], vy_back[-2 * i - 3], vz_back[-2 * i - 3] = v
mass_back[-2 * i - 3] = m
t_grid[-2 * i - 3] = t.start.mjd2000
x = x + x_back
y = y + y_back
z = z + z_back
vx = vx + vx_back
vy = vy + vy_back
vz = vz + vz_back
mass = mass + mass_back
return t_grid, list(zip(x, y, z)), list(zip(vx, vy, vz)), mass
def plot_leg(ax, leg, N=5, units=1, color='b', legend=False, plot_line = True):
"""
Plots a trajectory leg
USAGE: plot_leg(ax, leg, N=5, units=1, color='b', legend=False, no_trajectory = False):
* ax: 3D axis object created using fig.gca(projection='3d')
* leg: a PyKEP.sims_flanagan.leg
* N: number of points to be plotted along one arc
* units: the length unit to be used in the plot
* color: matplotlib color to use to plot the trajectory and the grid points
* legend when True it plots also the legend
* plot_line: when True plots also the trajectory (between mid-points and grid points)
EXAMPLE:
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
fig = plt.figure()
ax = fig.gca(projection='3d')
t1 = epoch(0)
pl = planet_ss('earth')
rE,vE = pl.eph(t1)
plot_planet(ax, pl,t0=t1, units=AU)
t2 = epoch(440)
pl = planet_ss('mars')
rM, vM = pl.eph(t2)
plot_planet(ax, pl,t0=t2, units=AU)
sc = sims_flanagan.spacecraft(4500,0.5,2500)
x0 = sims_flanagan.sc_state(rE,vE,sc.mass)
xe = sims_flanagan.sc_state(rM,vM,sc.mass)
l = sims_flanagan.leg(t1,x0,[1,0,0]*5,t2,xe,sc,MU_SUN)
plot_sf_leg(ax,l,units=AU)
"""
from PyKEP import propagate_lagrangian, AU, DAY2SEC, G0, propagate_taylor
import numpy as np
from scipy.linalg import norm
from math import exp
#We compute the number of segments for forward and backward propagation
n_seg = len(leg.get_throttles())
fwd_seg = (n_seg+1)/2
back_seg = n_seg/2
#We extract information on the spacecraft
sc = leg.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
#And on the leg
throttles = leg.get_throttles()
mu = leg.get_mu()
#Forward propagation
#x,y,z contain the cartesian components of all points containd in states
x = [0.0]*(fwd_seg+1)
y = [0.0]*(fwd_seg+1)
z = [0.0]*(fwd_seg+1)
state = leg.get_xi()
#Initial conditions
r = state.r; v = state.v; m = state.m
x[0] = r[0]/units
y[0] = r[1]/units
z[0] = r[2]/units
#We compute all points by propagation
for i,t in enumerate(throttles[:fwd_seg]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value),1.0)
#Taylor propagation of constant thrust u
u = [max_thrust * dumb for dumb in t.value]
if plot_line:
plot_taylor(ax,r,v,m,u,dt,mu,isp*G0,N=N,units=units,color=(alpha,0,1-alpha))
r,v,m = propagate_taylor(r,v,m,u,dt,mu,isp*G0,-10,-10)
x[i+1] = r[0]/units
y[i+1] = r[1]/units
z[i+1] = r[2]/units
ax.scatter(x[:-1], y[:-1], z[:-1], label='nodes',marker='o')
ax.scatter(x[-1],y[-1],z[-1], marker='^', c='y', label='mismatch point')
#Backward propagation
#x,y,z will contain the cartesian components of
x = [0.0]*(back_seg+1)
y = [0.0]*(back_seg+1)
z = [0.0]*(back_seg+1)
state = leg.get_xf()
#Final conditions
r = state.r; v = state.v; m = state.m
x[-1] = r[0]/units
y[-1] = r[1]/units
z[-1] = r[2]/units
for i,t in enumerate(throttles[-1:-back_seg-1:-1]):
dt = (t.end.mjd - t.start.mjd)*DAY2SEC
alpha = min(norm(t.value),1.0)
u = [max_thrust * dumb for dumb in t.value]
if plot_line:
plot_taylor(ax,r,v,m,u,-dt,mu,isp*G0,N=N,units=units,color=(alpha,0,1-alpha))
r,v,m = propagate_taylor(r,v,m,u,-dt,mu,isp*G0,-10,-10)
x[-i-2] = r[0]/units
y[-i-2] = r[1]/units
z[-i-2] = r[2]/units
ax.scatter(x[1:], y[1:], z[1:], marker = 'o')
ax.scatter(x[0], y[0], z[0], marker='^', c='y')
if legend:
ax.legend()
def point_sf_leg(leg, N=5, units=1, color='b'):
from PyKEP import propagate_lagrangian, AU, DAY2SEC, G0, propagate_taylor, SEC2DAY
import numpy as np
from scipy.linalg import norm
from math import exp, sqrt
# We compute the number of segments for forward and backward propagation
n_seg = len(leg.get_throttles())
fwd_seg = (n_seg + 1) // 2
back_seg = n_seg // 2
#we see if the propulsion is nuclear or solar powered
sp = leg.solar_powered;
# We extract information on the spacecraft
sc = leg.get_spacecraft()
isp = sc.isp
max_thrust = sc.thrust
# And on the leg
throttles = leg.get_throttles()
mu = leg.get_mu()
# Forward propagation
# x,y,z contain the cartesian components of all points (grid+midpints)
x_bounds = []
y_bounds = []
z_bounds = []
x = []
y = []
z = []
Tx = []
Ty = []
Tz = []
Tmax = []
mm = []
tt = []
state = leg.get_xi()
# Initial conditions
r = state.r
v = state.v
m = state.m
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
segmentID = 0
temptime = 0
# We compute all points by propagation
for i, t in enumerate(throttles[:fwd_seg]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
# Keplerian propagation and dV application
if leg.high_fidelity is False:
if(sp):
dSun = sqrt(r[0]*r[0]+r[1]*r[1]+r[2]*r[2])
max_thrust = sc.get_thrust_electricSolar(dSun)
isp = isp;
dV = [max_thrust / m * dt * dumb for dumb in t.value]
xi,yi,zi,ti = point_kepler(r, v, dt / 2, mu, N=N, units=units)
r, v = propagate_lagrangian(r, v, dt / 2, mu)
#x[-2 * i - 2] = r[0] / units
#y[-2 * i - 2] = r[1] / units
#z[-2 * i - 2] = r[2] / units
x = x + xi
y = y + yi
z = z + zi
tt = tt + [e + temptime for e in ti]
temptime = tt[-1]
mm = mm + [m]*len(ti)
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
# the thrust is constant on the segment (1st half-segment)
Tx = Tx + [t.value[0]]*len(ti)
Ty = Ty + [t.value[1]]*len(ti)
Tz = Tz + [t.value[2]]*len(ti)
Tmax = Tmax + [max_thrust]*len(ti)
# v= v+dV
v = [a + b for a, b in zip(v, dV)]
xi,yi,zi,ti = point_kepler(r, v, dt / 2, mu, N=N, units=units)
r, v = propagate_lagrangian(r, v, dt / 2, mu)
#x[-2 * i - 3] = r[0] / units
#y[-2 * i - 3] = r[1] / units
#z[-2 * i - 3] = r[2] / units
m *= exp(-norm(dV) / isp / G0)
x = x + xi
y = y + yi
z = z + zi
tt = tt + [e + temptime for e in ti]
temptime = tt[-1]
mm = mm+ [m]*len(ti)
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
# the thrust is constant on the segment (2nd half-segment)
Tx = Tx + [t.value[0]]*len(ti)
Ty = Ty + [t.value[1]]*len(ti)
Tz = Tz + [t.value[2]]*len(ti)
Tmax = Tmax + [max_thrust]*len(ti)
else:
if(sp):
dSun = sqrt(r[0]*r[0]+r[1]*r[1]+r[2]*r[2])
max_thrust = sc.get_thrust_electricSolar(dSun)
isp = isp;
u = [max_thrust * dumb for dumb in t.value]
xi,yi,zi,ti,mi = point_taylor(r, v, m, u, dt , mu, isp * G0, N=N, units=units)
x = x + xi
y = y + yi
z = z + zi
tt = tt + [e + temptime for e in ti]
temptime = tt[-1]
mm = mm + mi
r, v, m = propagate_taylor(r, v, m, u, dt , mu, isp * G0, -12, -12)
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
# the thrust is constant of the whole segment
Tx = Tx + [t.value[0]]*len(ti)
Ty = Ty + [t.value[1]]*len(ti)
Tz = Tz + [t.value[2]]*len(ti)
Tmax = Tmax + [max_thrust]*len(ti)
state = leg.get_xf()
# Final conditions
r = state.r
v = state.v
m = state.m
# used to reverse the order caused by the back-propagation
xt = []
yt = []
zt = []
mmt = []
ttt = []
Txt = []
Tyt = []
Tzt = []
Tmaxt = []
x_bounds = x_bounds + [r[0] / units] #gonna be the wrong side over...
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
temptime = 0
for i, t in enumerate(throttles[-1:-back_seg - 1:-1]):
dt = (t.end.mjd - t.start.mjd) * DAY2SEC
alpha = min(norm(t.value), 1.0)
if leg.high_fidelity is False:
if(sp):
dSun = sqrt(r[0]*r[0]+r[1]*r[1]+r[2]*r[2])
max_thrust = sc.get_thrust_electricSolar(dSun)
isp = isp;
dV = [max_thrust / m * dt * dumb for dumb in t.value]
xi,yi,zi,ti = point_kepler(r, v, -dt / 2, mu, N=N, units=units)
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
#x[-2 * i - 2] = r[0] / units
#y[-2 * i - 2] = r[1] / units
#z[-2 * i - 2] = r[2] / units
xt = xt + xi
yt = yt + yi
zt = zt + zi
ttt = ttt + [e + temptime for e in ti]
temptime = ttt[-1]
mmt = mmt + [m]*len(ti)
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
# the thrust is constant on the segment (1st half-segment)
Txt = Txt + [t.value[0]]*len(ti)
Tyt = Tyt + [t.value[1]]*len(ti)
Tzt = Tzt + [t.value[2]]*len(ti)
Tmaxt = Tmaxt + [max_thrust]*len(ti)
# v= v+dV
v = [a - b for a, b in zip(v, dV)]
xi,yi,zi,ti = point_kepler(r, v, -dt / 2, mu, N=N, units=units)
r, v = propagate_lagrangian(r, v, -dt / 2, mu)
#x[-2 * i - 3] = r[0] / units
#y[-2 * i - 3] = r[1] / units
#z[-2 * i - 3] = r[2] / units
m *= exp(norm(dV) / isp / G0)
xt = xt + xi
yt = yt + yi
zt = zt + zi
ttt = ttt + [e + temptime for e in ti]
temptime = ttt[-1]
mmt = mmt + [m]*len(ti)
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
# the thrust is constant on the segment (2nd half-segment)
Txt = Txt + [t.value[0]]*len(ti)
Tyt = Tyt + [t.value[1]]*len(ti)
Tzt = Tzt + [t.value[2]]*len(ti)
Tmaxt = Tmaxt + [max_thrust]*len(ti)
else:
if(sp):
dSun = sqrt(r[0]*r[0]+r[1]*r[1]+r[2]*r[2])
max_thrust = sc.get_thrust_electricSolar(dSun)
isp = isp;
u = [max_thrust * dumb for dumb in t.value]
xi,yi,zi,ti,mi = point_taylor(r, v, m, u, -dt , mu, isp * G0, N=N, units=units)
xt = xt + xi
yt = yt + yi
zt = zt + zi
ttt = ttt + [e + temptime for e in ti]
temptime = ttt[-1]
mmt = mmt + mi
r, v, m = propagate_taylor(r, v, m, u, -dt , mu, isp * G0, -12, -12)
x_bounds = x_bounds + [r[0] / units]
y_bounds = y_bounds + [r[1] / units]
z_bounds = z_bounds + [r[2] / units]
# the thrust is constant on the whole segment
Txt = Txt + [t.value[0]]*len(ti)
Tyt = Tyt + [t.value[1]]*len(ti)
Tzt = Tzt + [t.value[2]]*len(ti)
Tmaxt = Tmaxt + [max_thrust]*len(ti)
Tpx = Tx
Tpy = Ty
Tpz = Tz
x = x + xt[::-1]
y = y + yt[::-1]
z = z + zt[::-1]
Tx = Tx + Txt[::-1]
Ty = Ty + Tyt[::-1]
Tz = Tz + Tzt[::-1]
Tmax = Tmax + Tmaxt[::-1]
mm = mm + mmt[::-1]
temptime = tt[-1] - ttt[-1]
tt = tt + [e + temptime for e in ttt[::-1]]
return x,y,z,mm,tt,Tx,Ty,Tz,Tmax,x_bounds,y_bounds,z_bounds