Пример #1
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def test_atmospheric_drag():
    # http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node94.html#sair (10.148)
    # given the expression for \dot{r} / r, aproximate \Delta r \approx F_r * \Delta t

    R = Earth.R.to(u.km).value
    k = Earth.k.to(u.km**3 / u.s**2).value

    # parameters of a circular orbit with h = 250 km (any value would do, but not too small)
    orbit = Orbit.circular(Earth, 250 * u.km)
    r0, _ = orbit.rv()
    r0 = r0.to(u.km).value

    # parameters of a body
    C_D = 2.2  # dimentionless (any value would do)
    A = ((np.pi / 4.0) * (u.m**2)).to(u.km**2).value  # km^2
    m = 100  # kg
    B = C_D * A / m

    # parameters of the atmosphere
    rho0 = Earth.rho0.to(u.kg / u.km**3).value  # kg/km^3
    H0 = Earth.H0.to(u.km).value
    tof = 100000  # s

    dr_expected = -B * rho0 * np.exp(-(norm(r0) - R) / H0) * np.sqrt(k * norm(r0)) * tof
    # assuming the atmospheric decay during tof is small,
    # dr_expected = F_r * tof (Newton's integration formula), where
    # F_r = -B rho(r) |r|^2 sqrt(k / |r|^3) = -B rho(r) sqrt(k |r|)

    r, v = cowell(orbit, tof, ad=atmospheric_drag, R=R, C_D=C_D, A=A, m=m, H0=H0, rho0=rho0)

    assert_quantity_allclose(norm(r) - norm(r0), dr_expected, rtol=1e-2)
Пример #2
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def test_atmospheric_drag():
    # http://farside.ph.utexas.edu/teaching/celestial/Celestialhtml/node94.html#sair (10.148)
    # given the expression for \dot{r} / r, aproximate \Delta r \approx F_r * \Delta t

    R = Earth.R.to(u.km).value
    k = Earth.k.to(u.km**3 / u.s**2).value

    # parameters of a circular orbit with h = 250 km (any value would do, but not too small)
    orbit = Orbit.circular(Earth, 250 * u.km)
    r0, _ = orbit.rv()
    r0 = r0.to(u.km).value

    # parameters of a body
    C_D = 2.2  # dimentionless (any value would do)
    A = ((np.pi / 4.0) * (u.m**2)).to(u.km**2).value  # km^2
    m = 100  # kg
    B = C_D * A / m

    # parameters of the atmosphere
    rho0 = Earth.rho0.to(u.kg / u.km**3).value  # kg/km^3
    H0 = Earth.H0.to(u.km).value
    tof = 100000  # s

    dr_expected = -B * rho0 * np.exp(-(norm(r0) - R) / H0) * np.sqrt(k * norm(r0)) * tof
    # assuming the atmospheric decay during tof is small,
    # dr_expected = F_r * tof (Newton's integration formula), where
    # F_r = -B rho(r) |r|^2 sqrt(k / |r|^3) = -B rho(r) sqrt(k |r|)

    r, v = cowell(orbit, tof, ad=atmospheric_drag, R=R, C_D=C_D, A=A, m=m, H0=H0, rho0=rho0)

    assert_quantity_allclose(norm(r) - norm(r0), dr_expected, rtol=1e-2)
Пример #3
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def change_ecc_quasioptimal(ss_0, ecc_f, f):
    """Guidance law from the model.
    Thrust is aligned with an inertially fixed direction perpendicular to the
    semimajor axis of the orbit.

    Parameters
    ----------
    ss_0 : Orbit
        Initial orbit, containing all the information.
    ecc_f : float
        Final eccentricity.
    f : float
        Magnitude of constant acceleration
    """
    # We fix the inertial direction at the beginning
    k = ss_0.attractor.k.to(u.km**3 / u.s**2).value
    a = ss_0.a.to(u.km).value
    ecc_0 = ss_0.ecc.value
    if ecc_0 > 0.001:  # Arbitrary tolerance
        ref_vec = ss_0.e_vec / ecc_0
    else:
        ref_vec = ss_0.r / norm(ss_0.r)

    h_unit = ss_0.h_vec / norm(ss_0.h_vec)
    thrust_unit = cross(h_unit, ref_vec) * np.sign(ecc_f - ecc_0)

    def a_d(t0, u_, k):
        accel_v = f * thrust_unit
        return accel_v

    delta_V, t_f = extra_quantities(k, a, ecc_0, ecc_f, f)
    return a_d, delta_V, t_f
Пример #4
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def third_body(t0, state, k, k_third, third_body):
    r"""Calculates 3rd body acceleration (km/s2)

    .. math::

        \vec{p} = \mu_{m}\left ( \frac{\vec{r_{m/s}}}{r_{m/s}^3} - \frac{\vec{r_{m}}}{r_{m}^3} \right )

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    third_body: a callable object returning the position of 3rd body
        third body that causes the perturbation
    
    Note
    ----
    This formula is taken from Howard Curtis, section 12.10. As an example, a third body could be
    the gravity from the Moon acting on a small satellite.

    """

    body_r = third_body(t0)
    delta_r = body_r - state[:3]
    return k_third * delta_r / norm(delta_r)**3 - k_third * body_r / norm(
        body_r)**3
Пример #5
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def third_body(t0, state, k, k_third, third_body):
    r"""Calculates 3rd body acceleration (km/s2)

    .. math::

        \vec{p} = \mu_{m}\left ( \frac{\vec{r_{m/s}}}{r_{m/s}^3} - \frac{\vec{r_{m}}}{r_{m}^3} \right )

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    third_body: a callable object returning the position of 3rd body
        third body that causes the perturbation

    Note
    ----
    This formula is taken from Howard Curtis, section 12.10. As an example, a third body could be
    the gravity from the Moon acting on a small satellite.

    """

    body_r = third_body(t0)
    delta_r = body_r - state[:3]
    return k_third * delta_r / norm(delta_r) ** 3 - k_third * body_r / norm(body_r) ** 3
def change_inc_ecc(ss_0, ecc_f, inc_f, f):
    """Guidance law from the model.
    Thrust is aligned with an inertially fixed direction perpendicular to the
    semimajor axis of the orbit.

    Parameters
    ----------
    ss_0 : Orbit
        Initial orbit, containing all the information.
    ecc_f : float
        Final eccentricity.
    inc_f : float
        Final inclination.
    f : float
        Magnitude of constant acceleration.
    """

    # We fix the inertial direction at the beginning
    ecc_0 = ss_0.ecc.value
    if ecc_0 > 0.001:  # Arbitrary tolerance
        ref_vec = ss_0.e_vec / ecc_0
    else:
        ref_vec = ss_0.r / norm(ss_0.r)

    h_unit = ss_0.h_vec / norm(ss_0.h_vec)
    thrust_unit = cross(h_unit, ref_vec) * np.sign(ecc_f - ecc_0)

    inc_0 = ss_0.inc.to(u.rad).value
    argp = ss_0.argp.to(u.rad).value

    beta_0_ = beta(ecc_0, ecc_f, inc_0, inc_f, argp)

    def a_d(t0, u_, k):
        r = u_[:3]
        v = u_[3:]
        nu = rv2coe(k, r, v)[-1]
        beta_ = beta_0_ * np.sign(
            np.cos(nu)
        )  # The sign of ß reverses at minor axis crossings

        w_ = cross(r, v) / norm(cross(r, v))
        accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
        return accel_v

    delta_V, beta_, t_f = extra_quantities(
        ss_0.attractor.k.to(u.km ** 3 / u.s ** 2).value,
        ss_0.a.to(u.km).value,
        ecc_0,
        ecc_f,
        inc_0,
        inc_f,
        argp,
        f,
    )
    return a_d, delta_V, beta_, t_f
Пример #7
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def change_inc_ecc(ss_0, ecc_f, inc_f, f):
    """Guidance law from the model.
    Thrust is aligned with an inertially fixed direction perpendicular to the
    semimajor axis of the orbit.

    Parameters
    ----------
    ss_0 : Orbit
        Initial orbit, containing all the information.
    ecc_f : float
        Final eccentricity.
    inc_f : float
        Final inclination.
    f : float
        Magnitude of constant acceleration.
    """

    # We fix the inertial direction at the beginning
    ecc_0 = ss_0.ecc.value
    if ecc_0 > 0.001:  # Arbitrary tolerance
        ref_vec = ss_0.e_vec / ecc_0
    else:
        ref_vec = ss_0.r / norm(ss_0.r)

    h_unit = ss_0.h_vec / norm(ss_0.h_vec)
    thrust_unit = cross(h_unit, ref_vec) * np.sign(ecc_f - ecc_0)

    inc_0 = ss_0.inc.to(u.rad).value
    argp = ss_0.argp.to(u.rad).value

    beta_0_ = beta(ecc_0, ecc_f, inc_0, inc_f, argp)

    def a_d(t0, u_, k):
        r = u_[:3]
        v = u_[3:]
        nu = rv2coe(k, r, v)[-1]
        beta_ = beta_0_ * np.sign(
            np.cos(nu)
        )  # The sign of ß reverses at minor axis crossings

        w_ = cross(r, v) / norm(cross(r, v))
        accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
        return accel_v

    delta_V, beta_, t_f = extra_quantities(
        ss_0.attractor.k.to(u.km ** 3 / u.s ** 2).value,
        ss_0.a.to(u.km).value,
        ecc_0,
        ecc_f,
        inc_0,
        inc_f,
        argp,
        f,
    )
    return a_d, delta_V, beta_, t_f
Пример #8
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    def a_d(t0, u_, k):
        r = u_[:3]
        v = u_[3:]
        nu = rv2coe(k, r, v)[-1]

        alpha_ = nu - np.pi / 2

        r_ = r / norm(r)
        w_ = cross(r, v) / norm(cross(r, v))
        s_ = cross(w_, r_)
        accel_v = f * (np.cos(alpha_) * s_ + np.sin(alpha_) * r_)
        return accel_v
Пример #9
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    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
Пример #10
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    def a_d(t0, u_, k):
        r = u_[:3]
        v = u_[3:]
        nu = rv2coe(k, r, v)[-1]

        alpha_ = nu - np.pi / 2

        r_ = r / norm(r)
        w_ = cross(r, v) / norm(cross(r, v))
        s_ = cross(w_, r_)
        accel_v = f * (np.cos(alpha_) * s_ + np.sin(alpha_) * r_)
        return accel_v
Пример #11
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    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
Пример #12
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def J2_perturbation(t0, state, k, J2, R):
    """Calculates J2_perturbation acceleration (km/s2)

    .. versionadded:: 0.9.0

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    J2: float
        oblateness factor
    R: float
        attractor radius

    Notes
    -----
    The J2 accounts for the oblateness of the attractor. The formula is given in
    Howard Curtis, (12.30)

    """
    r_vec = state[:3]
    r = norm(r_vec)

    factor = (3.0 / 2.0) * k * J2 * (R**2) / (r**5)

    a_x = 5.0 * r_vec[2]**2 / r**2 - 1
    a_y = 5.0 * r_vec[2]**2 / r**2 - 1
    a_z = 5.0 * r_vec[2]**2 / r**2 - 3
    return np.array([a_x, a_y, a_z]) * r_vec * factor
Пример #13
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def J2_perturbation(t0, state, k, J2, R):
    """Calculates J2_perturbation acceleration (km/s2)

    .. versionadded:: 0.9.0

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    J2: float
        oblateness factor
    R: float
        attractor radius

    Notes
    -----
    The J2 accounts for the oblateness of the attractor. The formula is given in
    Howard Curtis, (12.30)

    """
    r_vec = state[:3]
    r = norm(r_vec)

    factor = (3.0 / 2.0) * k * J2 * (R ** 2) / (r ** 5)

    a_x = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
    a_y = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
    a_z = 5.0 * r_vec[2] ** 2 / r ** 2 - 3
    return np.array([a_x, a_y, a_z]) * r_vec * factor
Пример #14
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def J3_perturbation(t0, state, k, J3, R):
    """Calculates J3_perturbation acceleration (km/s2)

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    J3: float
        oblateness factor
    R: float
        attractor radius

    Notes
    -----
    The J3 accounts for the oblateness of the attractor. The formula is given in
    Howard Curtis, problem 12.8
    This perturbation has not been fully validated, see https://github.com/poliastro/poliastro/pull/398

    """
    r_vec = state[:3]
    r = norm(r_vec)

    factor = (1.0 / 2.0) * k * J3 * (R**3) / (r**5)
    cos_phi = r_vec[2] / r

    a_x = 5.0 * r_vec[0] / r * (7.0 * cos_phi**3 - 3.0 * cos_phi)
    a_y = 5.0 * r_vec[1] / r * (7.0 * cos_phi**3 - 3.0 * cos_phi)
    a_z = 3.0 * (35.0 / 3.0 * cos_phi**4 - 10.0 * cos_phi**2 + 1)
    return np.array([a_x, a_y, a_z]) * factor
Пример #15
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def J3_perturbation(t0, state, k, J3, R):
    """Calculates J3_perturbation acceleration (km/s2)

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    J3: float
        oblateness factor
    R: float
        attractor radius

    Notes
    -----
    The J3 accounts for the oblateness of the attractor. The formula is given in
    Howard Curtis, problem 12.8
    This perturbation has not been fully validated, see https://github.com/poliastro/poliastro/pull/398

    """
    r_vec = state[:3]
    r = norm(r_vec)

    factor = (1.0 / 2.0) * k * J3 * (R ** 3) / (r ** 5)
    cos_phi = r_vec[2] / r

    a_x = 5.0 * r_vec[0] / r * (7.0 * cos_phi ** 3 - 3.0 * cos_phi)
    a_y = 5.0 * r_vec[1] / r * (7.0 * cos_phi ** 3 - 3.0 * cos_phi)
    a_z = 3.0 * (35.0 / 3.0 * cos_phi ** 4 - 10.0 * cos_phi ** 2 + 1)
    return np.array([a_x, a_y, a_z]) * factor
Пример #16
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def atmospheric_drag(t0, state, k, R, C_D, A, m, H0, rho0):
    r"""Calculates atmospheric drag acceleration (km/s2)

    .. math::

        \vec{p} = -\frac{1}{2}\rho v_{rel}\left ( \frac{C_{d}A}{m} \right )\vec{v_{rel}}


    .. versionadded:: 0.9.0

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    R : float
        radius of the attractor (km)
    C_D: float
        dimensionless drag coefficient ()
    A: float
        frontal area of the spacecraft (km^2)
    m: float
        mass of the spacecraft (kg)
    H0 : float
        atmospheric scale height, (km)
    rho0: float
        the exponent density pre-factor, (kg / m^3)

    Note
    ----
    This function provides the acceleration due to atmospheric drag. We follow
    Howard Curtis, section 12.4
    the atmospheric density model is rho(H) = rho0 x exp(-H / H0)

    """
    H = norm(state[:3])

    v_vec = state[3:]
    v = norm(v_vec)
    B = C_D * A / m
    rho = rho0 * np.exp(-(H - R) / H0)

    return -(1.0 / 2.0) * rho * B * v * v_vec
Пример #17
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def third_body(t0, state, k, k_third, third_body):
    """Calculates 3rd body acceleration (km/s2)

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    third_body: a callable object returning the position of 3rd body
        third body that causes the perturbation
    """

    body_r = third_body(t0)
    delta_r = body_r - state[:3]
    return k_third * delta_r / norm(delta_r) ** 3 - k_third * body_r / norm(body_r) ** 3
Пример #18
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def atmospheric_drag(t0, state, k, R, C_D, A, m, H0, rho0):
    r"""Calculates atmospheric drag acceleration (km/s2)

    .. math::

        \vec{p} = -\frac{1}{2}\rho v_{rel}\left ( \frac{C_{d}A}{m} \right )\vec{v_{rel}}


    .. versionadded:: 0.9.0

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    C_D: float
        dimensionless drag coefficient ()
    A: float
        frontal area of the spacecraft (km^2)
    m: float
        mass of the spacecraft (kg)
    H0 : float
        atmospheric scale height, (km)
    rho0: float
        the exponent density pre-factor, (kg / m^3)

    Note
    ----
    This function provides the acceleration due to atmospheric drag. We follow
    Howard Curtis, section 12.4
    the atmospheric density model is rho(H) = rho0 x exp(-H / H0)

    """
    H = norm(state[:3])

    v_vec = state[3:]
    v = norm(v_vec)
    B = C_D * A / m
    rho = rho0 * np.exp(-(H - R) / H0)

    return -(1.0 / 2.0) * rho * B * v * v_vec
Пример #19
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def izzo(k, r1, r2, tof, M, numiter, rtol):
    # Check preconditions
    assert tof > 0
    assert k > 0

    # Chord
    c = r2 - r1
    c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)

    # Semiperimeter
    s = (r1_norm + r2_norm + c_norm) * .5

    # Versors
    i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
    i_h = cross(i_r1, i_r2)
    i_h = i_h / norm(i_h)  # Fixed from paper

    # Geometry of the problem
    ll = np.sqrt(1 - c_norm / s)

    if i_h[2] < 0:
        ll = -ll
        i_h = -i_h

    i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2)  # Fixed from paper

    # Non dimensional time of flight
    T = np.sqrt(2 * k / s ** 3) * tof

    # Find solutions
    xy = _find_xy(ll, T, M, numiter, rtol)

    # Reconstruct
    gamma = np.sqrt(k * s / 2)
    rho = (r1_norm - r2_norm) / c_norm
    sigma = np.sqrt(1 - rho ** 2)

    for x, y in xy:
        V_r1, V_r2, V_t1, V_t2 = _reconstruct(x, y, r1_norm, r2_norm, ll,
                                              gamma, rho, sigma)
        v1 = V_r1 * i_r1 + V_t1 * i_t1
        v2 = V_r2 * i_r2 + V_t2 * i_t2
        yield v1, v2
Пример #20
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def third_body(t0, state, k, k_third, third_body):
    """Calculates 3rd body acceleration (km/s2)

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    third_body: a callable object returning the position of 3rd body
        third body that causes the perturbation
    """

    body_r = third_body(t0)
    delta_r = body_r - state[:3]
    return k_third * delta_r / norm(delta_r)**3 - k_third * body_r / norm(
        body_r)**3
Пример #21
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def izzo(k, r1, r2, tof, M, numiter, rtol):
    # Check preconditions
    assert tof > 0
    assert k > 0

    # Chord
    c = r2 - r1
    c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)

    # Semiperimeter
    s = (r1_norm + r2_norm + c_norm) * .5

    # Versors
    i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
    i_h = cross(i_r1, i_r2)
    i_h = i_h / norm(i_h)  # Fixed from paper

    # Geometry of the problem
    ll = np.sqrt(1 - c_norm / s)

    if i_h[2] < 0:
        ll = -ll
        i_h = -i_h

    i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2)  # Fixed from paper

    # Non dimensional time of flight
    T = np.sqrt(2 * k / s**3) * tof

    # Find solutions
    xy = _find_xy(ll, T, M, numiter, rtol)

    # Reconstruct
    gamma = np.sqrt(k * s / 2)
    rho = (r1_norm - r2_norm) / c_norm
    sigma = np.sqrt(1 - rho**2)

    for x, y in xy:
        V_r1, V_r2, V_t1, V_t2 = _reconstruct(x, y, r1_norm, r2_norm, ll,
                                              gamma, rho, sigma)
        v1 = V_r1 * i_r1 + V_t1 * i_t1
        v2 = V_r2 * i_r2 + V_t2 * i_t2
        yield v1, v2
Пример #22
0
    def a_d(t0, u_, k):
        r = u_[:3]
        v = u_[3:]
        nu = rv2coe(k, r, v)[-1]
        beta_ = beta_0_ * np.sign(
            np.cos(nu))  # The sign of ß reverses at minor axis crossings

        w_ = cross(r, v) / norm(cross(r, v))
        accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
        return accel_v
Пример #23
0
    def a_d(t0, u_, k):
        r = u_[:3]
        v = u_[3:]
        nu = rv2coe(k, r, v)[-1]
        beta_ = beta_0_ * np.sign(
            np.cos(nu)
        )  # The sign of ß reverses at minor axis crossings

        w_ = cross(r, v) / norm(cross(r, v))
        accel_v = f * (np.cos(beta_) * thrust_unit + np.sin(beta_) * w_)
        return accel_v
Пример #24
0
def radiation_pressure(t0, state, k, R, C_R, A, m, Wdivc_s, star):
    r"""Calculates radiation pressure acceleration (km/s2)

    .. math::

        \vec{p} = -\nu \frac{S}{c} \left ( \frac{C_{r}A}{m} \right )\frac{\vec{r}}{r}

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    R : float
        radius of the attractor
    C_R: float
        dimensionless radiation pressure coefficient, 1 < C_R < 2 ()
    A: float
        effective spacecraft area (km^2)
    m: float
        mass of the spacecraft (kg)
    Wdivc_s : float
        total star emitted power divided by the speed of light (W * s / km)
    star: a callable object returning the position of star in attractor frame
        star position

    Note
    ----
    This function provides the acceleration due to star light pressure. We follow
    Howard Curtis, section 12.9

    """

    r_star = star(t0)
    r_sat = state[:3]
    P_s = Wdivc_s / (norm(r_star) ** 2)

    nu = float(shadow_function(r_sat, r_star, R))
    return -nu * P_s * (C_R * A / m) * r_star / norm(r_star)
Пример #25
0
def radiation_pressure(t0, state, k, R, C_R, A, m, Wdivc_s, star):
    r"""Calculates radiation pressure acceleration (km/s2)

    .. math::

        \vec{p} = -\nu \frac{S}{c} \left ( \frac{C_{r}A}{m} \right )\frac{\vec{r}}{r}

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    R : float
        radius of the attractor
    C_R: float
        dimensionless radiation pressure coefficient, 1 < C_R < 2 ()
    A: float
        effective spacecraft area (km^2)
    m: float
        mass of the spacecraft (kg)
    Wdivc_s : float
        total star emitted power divided by the speed of light (W * s / km)
    star: a callable object returning the position of star in attractor frame
        star position
    
    Note
    ----
    This function provides the acceleration due to star light pressure. We follow
    Howard Curtis, section 12.9

    """

    r_star = star(t0)
    r_sat = state[:3]
    P_s = Wdivc_s / (norm(r_star)**2)

    nu = float(shadow_function(r_sat, r_star, R))
    return -nu * P_s * (C_R * A / m) * r_star / norm(r_star)
Пример #26
0
def rv2coe(k, r, v, tol=1e-8):
    """Converts from vectors to classical orbital elements.

    Parameters
    ----------
    k : float
        Standard gravitational parameter (km^3 / s^2).
    r : array
        Position vector (km).
    v : array
        Velocity vector (km / s).
    tol : float, optional
        Tolerance for eccentricity and inclination checks, default to 1e-8.

    """
    h = cross(r, v)
    n = cross([0, 0, 1], h) / norm(h)
    e = ((v.dot(v) - k / (norm(r))) * r - r.dot(v) * v) / k
    ecc = norm(e)
    p = h.dot(h) / k
    inc = np.arccos(h[2] / norm(h))

    circular = ecc < tol
    equatorial = abs(inc) < tol

    if equatorial and not circular:
        raan = 0
        argp = np.arctan2(e[1], e[0]) % (2 * np.pi)  # Longitude of periapsis
        nu = (np.arctan2(h.dot(cross(e, r)) / norm(h), r.dot(e)) %
              (2 * np.pi))
    elif not equatorial and circular:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = 0
        # Argument of latitude
        nu = (np.arctan2(r.dot(cross(h, n)) / norm(h), r.dot(n)) %
              (2 * np.pi))
    elif equatorial and circular:
        raan = 0
        argp = 0
        nu = np.arctan2(r[1], r[0]) % (2 * np.pi)  # True longitude
    else:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = (np.arctan2(e.dot(cross(h, n)) / norm(h), e.dot(n)) %
                (2 * np.pi))
        nu = (np.arctan2(r.dot(cross(h, e)) / norm(h), r.dot(e))
              % (2 * np.pi))

    return p, ecc, inc, raan, argp, nu
Пример #27
0
def rv2coe(k, r, v, tol=1e-8):
    """Converts from vectors to classical orbital elements.

    Parameters
    ----------
    k : float
        Standard gravitational parameter (km^3 / s^2).
    r : array
        Position vector (km).
    v : array
        Velocity vector (km / s).
    tol : float, optional
        Tolerance for eccentricity and inclination checks, default to 1e-8.

    """
    h = cross(r, v)
    n = cross([0, 0, 1], h) / norm(h)
    e = ((v.dot(v) - k / (norm(r))) * r - r.dot(v) * v) / k
    ecc = norm(e)
    p = h.dot(h) / k
    inc = np.arccos(h[2] / norm(h))

    circular = ecc < tol
    equatorial = abs(inc) < tol

    if equatorial and not circular:
        raan = 0
        argp = np.arctan2(e[1], e[0]) % (2 * np.pi)  # Longitude of periapsis
        nu = (np.arctan2(h.dot(cross(e, r)) / norm(h), r.dot(e)) % (2 * np.pi))
    elif not equatorial and circular:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = 0
        # Argument of latitude
        nu = (np.arctan2(r.dot(cross(h, n)) / norm(h), r.dot(n)) % (2 * np.pi))
    elif equatorial and circular:
        raan = 0
        argp = 0
        nu = np.arctan2(r[1], r[0]) % (2 * np.pi)  # True longitude
    else:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = (np.arctan2(e.dot(cross(h, n)) / norm(h), e.dot(n)) %
                (2 * np.pi))
        nu = (np.arctan2(r.dot(cross(h, e)) / norm(h), r.dot(e)) % (2 * np.pi))

    return p, ecc, inc, raan, argp, nu
Пример #28
0
def J2_perturbation(t0, state, k, J2, R):
    r"""Calculates J2_perturbation acceleration (km/s2)

    .. math::

        \vec{p} = \frac{3}{2}\frac{J_{2}\mu R^{2}}{r^{4}}\left [\frac{x}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{i} + \frac{y}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{j} + \frac{z}{r}\left ( 5\frac{z^{2}}{r^{2}}-3 \right )\vec{k}\right]

    .. versionadded:: 0.9.0

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    J2: float
        oblateness factor
    R: float
        attractor radius

    Note
    ----
    The J2 accounts for the oblateness of the attractor. The formula is given in
    Howard Curtis, (12.30)

    """
    r_vec = state[:3]
    r = norm(r_vec)

    factor = (3.0 / 2.0) * k * J2 * (R ** 2) / (r ** 5)

    a_x = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
    a_y = 5.0 * r_vec[2] ** 2 / r ** 2 - 1
    a_z = 5.0 * r_vec[2] ** 2 / r ** 2 - 3
    return np.array([a_x, a_y, a_z]) * r_vec * factor
Пример #29
0
def J2_perturbation(t0, state, k, J2, R):
    r"""Calculates J2_perturbation acceleration (km/s2)

    .. math::

        \vec{p} = \frac{3}{2}\frac{J_{2}\mu R^{2}}{r^{4}}\left [\frac{x}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{i} + \frac{y}{r}\left ( 5\frac{z^{2}}{r^{2}}-1 \right )\vec{j} + \frac{z}{r}\left ( 5\frac{z^{2}}{r^{2}}-3 \right )\vec{k}\right]

    .. versionadded:: 0.9.0

    Parameters
    ----------
    t0 : float
        Current time (s)
    state : numpy.ndarray
        Six component state vector [x, y, z, vx, vy, vz] (km, km/s).
    k : float
        gravitational constant, (km^3/s^2)
    J2: float
        oblateness factor
    R: float
        attractor radius

    Note
    ----
    The J2 accounts for the oblateness of the attractor. The formula is given in
    Howard Curtis, (12.30)

    """
    r_vec = state[:3]
    r = norm(r_vec)

    factor = (3.0 / 2.0) * k * J2 * (R**2) / (r**5)

    a_x = 5.0 * r_vec[2]**2 / r**2 - 1
    a_y = 5.0 * r_vec[2]**2 / r**2 - 1
    a_z = 5.0 * r_vec[2]**2 / r**2 - 3
    return np.array([a_x, a_y, a_z]) * r_vec * factor
Пример #30
0
def normalize_to_Curtis(t0, sun_r):
    r = sun_r(t0)
    return 149600000 * r / norm(r)
Пример #31
0
def rv2coe(k, r, v, tol=1e-8):
    r"""Converts from vectors to classical orbital elements.

    1. First the angular momentum is computed:
        .. math::
            \vec{h} = \vec{r} \times \vec{v}

    2. With it the eccentricity can be solved:
        .. math::
            \begin{align}
            \vec{e} &= \frac{1}{\mu}\left [ \left ( v^{2} - \frac{\mu}{r}\right ) \vec{r}  - (\vec{r} \cdot \vec{v})\vec{v} \right ] \\
            e &= \sqrt{\vec{e}\cdot\vec{e}} \\
            \end{align}

    3. The node vector line is solved:
        .. math::
            \begin{align}
            \vec{N} &= \vec{k} \times \vec{h} \\
            N &= \sqrt{\vec{N}\cdot\vec{N}}
            \end{align}

    4. The rigth ascension node is computed:
        .. math::
            \Omega = \left\{ \begin{array}{lcc}
             cos^{-1}{\left ( \frac{N_{x}}{N} \right )} &   if  & N_{y} \geq  0 \\
             \\ 360^{o} -cos^{-1}{\left ( \frac{N_{x}}{N} \right )} &  if & N_{y} < 0 \\
             \end{array}
            \right.

    5. The argument of perigee:
        .. math::
            \omega  = \left\{ \begin{array}{lcc}
             cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} &   if  & e_{z} \geq  0 \\
             \\ 360^{o} -cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} &  if & e_{z} < 0 \\
             \end{array}
            \right.

    6. And finally the true anomaly:
        .. math::
            \nu  = \left\{ \begin{array}{lcc}
             cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} &   if  & v_{r} \geq  0 \\
             \\ 360^{o} -cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} &  if & v_{r} < 0 \\
             \end{array}
            \right.

    Parameters
    ----------
    k : float
        Standard gravitational parameter (km^3 / s^2)
    r : array
        Position vector (km)
    v : array
        Velocity vector (km / s)
    tol : float, optional
        Tolerance for eccentricity and inclination checks, default to 1e-8

    Returns
    -------
    p : float
        Semi-latus rectum of parameter (km)
    ecc: float
        Eccentricity
    inc: float
        Inclination (rad)
    raan: float
        Right ascension of the ascending nod (rad)
    argp: float
        Argument of Perigee (rad)
    nu: float
        True Anomaly (rad)

    Examples
    --------
    >>> from poliastro.constants import GM_earth
    >>> from astropy import units as u
    >>> k = GM_earth.to(u.km ** 3 / u.s ** 2).value  # Earth gravitational parameter
    >>> r = np.array([-6045., -3490., 2500.])
    >>> v = np.array([-3.457, 6.618, 2.533])
    >>> p, ecc, inc, raan, argp, nu = rv2coe(k, r, v)
    >>> print("p:", p, "[km]")
    p: 8530.47436396927 [km]
    >>> print("ecc:", ecc)
    ecc: 0.17121118195416898
    >>> print("inc:", np.rad2deg(inc), "[deg]")
    inc: 153.2492285182475 [deg]
    >>> print("raan:", np.rad2deg(raan), "[deg]")
    raan: 255.27928533439618 [deg]
    >>> print("argp:", np.rad2deg(argp), "[deg]")
    argp: 20.068139973005366 [deg]
    >>> print("nu:", np.rad2deg(nu), "[deg]")
    nu: 28.445804984192122 [deg]

    Note
    ----
    This example is a real exercise from Orbital Mechanics for Engineering
    students by Howard D.Curtis. This exercise is 4.3 of 3rd. Edition, page 200.
    """

    h = cross(r, v)
    n = cross([0, 0, 1], h)
    e = ((v.dot(v) - k / (norm(r))) * r - r.dot(v) * v) / k
    ecc = norm(e)
    p = h.dot(h) / k
    inc = np.arccos(h[2] / norm(h))

    circular = ecc < tol
    equatorial = abs(inc) < tol

    if equatorial and not circular:
        raan = 0
        argp = np.arctan2(e[1], e[0]) % (2 * np.pi)  # Longitude of periapsis
        nu = np.arctan2(h.dot(cross(e, r)) / norm(h), r.dot(e))
    elif not equatorial and circular:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = 0
        # Argument of latitude
        nu = np.arctan2(r.dot(cross(h, n)) / norm(h), r.dot(n))
    elif equatorial and circular:
        raan = 0
        argp = 0
        nu = np.arctan2(r[1], r[0]) % (2 * np.pi)  # True longitude
    else:
        a = p / (1 - (ecc ** 2))
        ka = k * a
        if a > 0:
            e_se = r.dot(v) / sqrt(ka)
            e_ce = norm(r) * (norm(v) ** 2) / k - 1
            nu = E_to_nu(np.arctan2(e_se, e_ce), ecc)
        else:
            e_sh = r.dot(v) / sqrt(-ka)
            e_ch = norm(r) * (norm(v) ** 2) / k - 1
            nu = F_to_nu(np.log((e_ch + e_sh) / (e_ch - e_sh)) / 2, ecc)

        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        px = r.dot(n)
        py = r.dot(cross(h, n)) / norm(h)
        argp = (np.arctan2(py, px) - nu) % (2 * np.pi)

    nu = (nu + np.pi) % (2 * np.pi) - np.pi

    return p, ecc, inc, raan, argp, nu
Пример #32
0
def rv2coe(k, r, v, tol=1e-8):
    r"""Converts from vectors to classical orbital elements.

    1. First the angular momentum is computed:
        .. math::
            \vec{h} = \vec{r} \times \vec{v}

    2. With it the eccentricity can be solved:
        .. math::
            \begin{align} 
            \vec{e} &= \frac{1}{\mu}\left [ \left ( v^{2} - \frac{\mu}{r}\right ) \vec{r}  - (\vec{r} \cdot \vec{v})\vec{v} \right ] \\
            e &= \sqrt{\vec{e}\cdot\vec{e}} \\
            \end{align}
    
    3. The node vector line is solved:
        .. math::
            \begin{align}
            \vec{N} &= \vec{k} \times \vec{h} \\
            N &= \sqrt{\vec{N}\cdot\vec{N}}
            \end{align}

    4. The rigth ascension node is computed:
        .. math::
            \Omega = \left\{ \begin{array}{lcc}
             cos^{-1}{\left ( \frac{N_{x}}{N} \right )} &   if  & N_{y} \geq  0 \\
             \\ 360^{o} -cos^{-1}{\left ( \frac{N_{x}}{N} \right )} &  if & N_{y} < 0 \\
             \end{array}
            \right.

    5. The argument of perigee:
        .. math::
            \omega  = \left\{ \begin{array}{lcc}
             cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} &   if  & e_{z} \geq  0 \\
             \\ 360^{o} -cos^{-1}{\left ( \frac{\vec{N}\vec{e}}{Ne} \right )} &  if & e_{z} < 0 \\
             \end{array}
            \right.

    6. And finally the true anomaly:
        .. math::
            \nu  = \left\{ \begin{array}{lcc}
             cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} &   if  & v_{r} \geq  0 \\
             \\ 360^{o} -cos^{-1}{\left ( \frac{\vec{e}\vec{r}}{er} \right )} &  if & v_{r} < 0 \\
             \end{array}
            \right.

    Parameters
    ----------
    k : float
        Standard gravitational parameter (km^3 / s^2)
    r : array
        Position vector (km)
    v : array
        Velocity vector (km / s)
    tol : float, optional
        Tolerance for eccentricity and inclination checks, default to 1e-8

    Returns
    -------

    p : float
        Semi-latus rectum of parameter (km)
    ecc: float
        Eccentricity
    inc: float
        Inclination (rad)
    raan: float
        Right ascension of the ascending nod (rad)
    argp: float
        Argument of Perigee (rad)
    nu: float
        True Anomaly (rad)

    Examples
    --------
    >>> from poliastro.core.elements import rv2coe
    >>> from poliastro.constants import GM_earth
    >>> from astropy import units as u
    >>> import numpy as np
    >>> k = GM_earth.to(u.km**3 / u.s**2) #Earth gravitational parameter
    >>> r = [-6045, -3490, 2500] * u.km
    >>> v = [-3.457, 6.618, 2.533] * u.km / u.s
    >>> p, ecc, inc, raan, argp, nu = rv2coe(k, r, v)
    >>> print("p:", p, "[km]")
    p: 8530.47436396927 [km]
    >>> print("ecc:", ecc)
    ecc: 0.17121118195416898
    >>> print("inc:", np.rad2deg(inc), "[deg]")
    inc: 153.2492285182475 [deg]
    >>> print("raan:", np.rad2deg(raan), "[deg]")
    raan: 255.27928533439618 [deg]
    >>> print("argp:", np.rad2deg(argp), "[deg]")
    argp: 20.068139973005362 [deg]
    >>> print("nu:", np.rad2deg(nu), "[deg]")
    nu: 28.445804984192108 [deg]


    Note
    ----

    This example is a real exercise from Orbital Mechanics for Engineering
    students by Howard D.Curtis. This exercise is 4.3 of 3rd. Edition, page 200.
    """

    h = cross(r, v)
    n = cross([0, 0, 1], h) / norm(h)
    e = ((v.dot(v) - k / (norm(r))) * r - r.dot(v) * v) / k
    ecc = norm(e)
    p = h.dot(h) / k
    inc = np.arccos(h[2] / norm(h))

    circular = ecc < tol
    equatorial = abs(inc) < tol

    if equatorial and not circular:
        raan = 0
        argp = np.arctan2(e[1], e[0]) % (2 * np.pi)  # Longitude of periapsis
        nu = np.arctan2(h.dot(cross(e, r)) / norm(h), r.dot(e)) % (2 * np.pi)
    elif not equatorial and circular:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = 0
        # Argument of latitude
        nu = np.arctan2(r.dot(cross(h, n)) / norm(h), r.dot(n)) % (2 * np.pi)
    elif equatorial and circular:
        raan = 0
        argp = 0
        nu = np.arctan2(r[1], r[0]) % (2 * np.pi)  # True longitude
    else:
        raan = np.arctan2(n[1], n[0]) % (2 * np.pi)
        argp = np.arctan2(e.dot(cross(h, n)) / norm(h), e.dot(n)) % (2 * np.pi)
        nu = np.arctan2(r.dot(cross(h, e)) / norm(h), r.dot(e)) % (2 * np.pi)

    return p, ecc, inc, raan, argp, nu
Пример #33
0
def izzo(k, r1, r2, tof, M, numiter, rtol):
    """ Aplies izzo algorithm to solve Lambert's problem.

    Parameters
    ----------
    k: float
        Gravitational Constant
    r1: ~numpy.array 
        Initial position vector
    r2: ~numpy.array
        Final position vector
    tof: float
        Time of flight between both positions
    M: int
        Number of revolutions
    numiter: int
        Numbert of iterations
    rotl: float
        Error tolerance
    
    Returns
    -------

    v1: ~numpy.array
        Initial velocity vector
    v2: ~numpy.array
        FInal velocity vector

    """

    # Check preconditions
    assert tof > 0
    assert k > 0

    # Check collinearity of r1 and r2
    if np.all(cross(r1, r2) == 0):
        raise ValueError(
            "Lambert solution cannot be computed for collinear vectors")

    # Chord
    c = r2 - r1
    c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)

    # Semiperimeter
    s = (r1_norm + r2_norm + c_norm) * 0.5

    # Versors
    i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
    i_h = cross(i_r1, i_r2)
    i_h = i_h / norm(i_h)  # Fixed from paper

    # Geometry of the problem
    ll = np.sqrt(1 - min(1.0, c_norm / s))

    if i_h[2] < 0:
        ll = -ll
        i_h = -i_h

    i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2)  # Fixed from paper

    # Non dimensional time of flight
    T = np.sqrt(2 * k / s**3) * tof

    # Find solutions
    xy = _find_xy(ll, T, M, numiter, rtol)

    # Reconstruct
    gamma = np.sqrt(k * s / 2)
    rho = (r1_norm - r2_norm) / c_norm
    sigma = np.sqrt(1 - rho**2)

    for x, y in xy:
        V_r1, V_r2, V_t1, V_t2 = _reconstruct(x, y, r1_norm, r2_norm, ll,
                                              gamma, rho, sigma)
        v1 = V_r1 * i_r1 + V_t1 * i_t1
        v2 = V_r2 * i_r2 + V_t2 * i_t2
        yield v1, v2
Пример #34
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def normalize_to_Curtis(t0, sun_r):
    r = sun_r(t0)
    return 149600000 * r / norm(r)
Пример #35
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def izzo(k, r1, r2, tof, M, numiter, rtol):
    """ Aplies izzo algorithm to solve Lambert's problem.

    Parameters
    ----------
    k: float
        Gravitational Constant
    r1: ~numpy.array
        Initial position vector
    r2: ~numpy.array
        Final position vector
    tof: float
        Time of flight between both positions
    M: int
        Number of revolutions
    numiter: int
        Numbert of iterations
    rtol: float
        Error tolerance

    Returns
    -------

    v1: ~numpy.array
        Initial velocity vector
    v2: ~numpy.array
        Final velocity vector

    """

    # Check preconditions
    assert tof > 0
    assert k > 0

    # Check collinearity of r1 and r2
    if np.all(cross(r1, r2) == 0):
        raise ValueError("Lambert solution cannot be computed for collinear vectors")

    # Chord
    c = r2 - r1
    c_norm, r1_norm, r2_norm = norm(c), norm(r1), norm(r2)

    # Semiperimeter
    s = (r1_norm + r2_norm + c_norm) * 0.5

    # Versors
    i_r1, i_r2 = r1 / r1_norm, r2 / r2_norm
    i_h = cross(i_r1, i_r2)
    i_h = i_h / norm(i_h)  # Fixed from paper

    # Geometry of the problem
    ll = np.sqrt(1 - min(1.0, c_norm / s))

    if i_h[2] < 0:
        ll = -ll
        i_h = -i_h

    i_t1, i_t2 = cross(i_h, i_r1), cross(i_h, i_r2)  # Fixed from paper

    # Non dimensional time of flight
    T = np.sqrt(2 * k / s ** 3) * tof

    # Find solutions
    xy = _find_xy(ll, T, M, numiter, rtol)

    # Reconstruct
    gamma = np.sqrt(k * s / 2)
    rho = (r1_norm - r2_norm) / c_norm
    sigma = np.sqrt(1 - rho ** 2)

    for x, y in xy:
        V_r1, V_r2, V_t1, V_t2 = _reconstruct(
            x, y, r1_norm, r2_norm, ll, gamma, rho, sigma
        )
        v1 = V_r1 * i_r1 + V_t1 * i_t1
        v2 = V_r2 * i_r2 + V_t2 * i_t2
        yield v1, v2