def change_a_inc(k, a_0, a_f, inc_0, inc_f, f):
"""Guidance law from the Edelbaum/Kéchichian theory, optimal transfer between circular inclined orbits
(a_0, i_0) --> (a_f, i_f), ecc = 0.
Parameters
----------
k : float
Gravitational parameter.
a_0 : float
Initial semimajor axis.
a_f : float
Final semimajor axis.
inc_0 : float
Initial inclination.
inc_f : float
Final inclination.
f : float
Magnitude of constant acceleration
Notes
-----
Edelbaum theory, reformulated by Kéchichian.
References
----------
* Edelbaum, T. N. "Propulsion Requirements for Controllable
Satellites", 1961.
* Kéchichian, J. A. "Reformulation of Edelbaum's Low-Thrust
Transfer Problem Using Optimal Control Theory", 1997.
"""
V_0, beta_0_, _ = compute_parameters(k, a_0, a_f, inc_0, inc_f)
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
# Change sign of beta with the out-of-plane velocity
beta_ = beta(t0, V_0=V_0, f=f, beta_0=beta_0_) * np.sign(r[0] * (inc_f - inc_0))
t_ = v / norm(v)
w_ = cross(r, v) / norm(cross(r, v))
# n_ = cross(t_, w_)
accel_v = f * (
np.cos(beta_) * t_ +
np.sin(beta_) * w_
)
return accel_v
delta_V, t_f = extra_quantities(k, a_0, a_f, inc_0, inc_f, f)
return a_d, delta_V, t_f
def change_a_inc(k, a_0, a_f, inc_0, inc_f, f):
"""Guidance law from the Edelbaum/Kéchichian theory, optimal transfer between circular inclined orbits
(a_0, i_0) --> (a_f, i_f), ecc = 0.
Parameters
----------
k : float
Gravitational parameter.
a_0 : float
Initial semimajor axis.
a_f : float
Final semimajor axis.
inc_0 : float
Initial inclination.
inc_f : float
Final inclination.
f : float
Magnitude of constant acceleration
Notes
-----
Edelbaum theory, reformulated by Kéchichian.
References
----------
* Edelbaum, T. N. "Propulsion Requirements for Controllable
Satellites", 1961.
* Kéchichian, J. A. "Reformulation of Edelbaum's Low-Thrust
Transfer Problem Using Optimal Control Theory", 1997.
"""
V_0, beta_0_, _ = compute_parameters(k, a_0, a_f, inc_0, inc_f)
def a_d(t0, u_, k):
r = u_[:3]
v = u_[3:]
# Change sign of beta with the out-of-plane velocity
beta_ = beta(t0, V_0=V_0, f=f, beta_0=beta_0_) * np.sign(r[0] * (inc_f - inc_0))
t_ = v / norm(v)
w_ = cross(r, v) / norm(cross(r, v))
# n_ = cross(t_, w_)
accel_v = f * (np.cos(beta_) * t_ + np.sin(beta_) * w_)
return accel_v
delta_V, t_f = extra_quantities(k, a_0, a_f, inc_0, inc_f, f)
return a_d, delta_V, t_f