NP-C Problem import numpy as np from scipy.optimize import curve_fit
def smooth_attitude_interpolation(Cs, Cf, ωs, ωf, T): """ Smoothly interpolates between two attitude matrices Cs and Cf. The angular velocity and acceleration are continuous, and the jerk is continuous.
Args:
Cs: The initial attitude matrix.
Cf: The final attitude matrix.
ωs: The initial angular velocity.
ωf: The final angular velocity.
T: The time interval between Cs and Cf.
Returns:
A list of attitude matrices that interpolate between Cs and Cf.
"""
# Fit a cubic polynomial to the rotation vector.
θ = np.linspace(0, T, 3)
def rotation_vector(t):
return np.log(Cs.T @ Cf)
θ_poly, _ = curve_fit(rotation_vector, θ, np.zeros_like(θ), maxfev=100000)
# Compute the angular velocity and acceleration from the rotation vector polynomial.
ω = np.diff(θ_poly) / θ
ω_̇ = np.diff(ω) / θ
# Set the jerk at the endpoints to be equal to each other.
ω_̇[0] = ω_̇[-1]
# Solve for the angular velocities.
ω = np.linalg.solve(np.diag(1 / θ) + np.diag(ω_̇), ωs - ωf)
# Interpolate the attitude matrices.
C = np.exp(θ_poly @ np.linalg.inv(np.diag(θ)))
# Adjust the attitude matrices to account for time travel.
C = C * np.exp(-1j * 2 * np.pi * T)
return C
def solve_np_c(Cs, Cf, ωs, ωf, T): """ Solves the NP-C problem for the given initial and final attitude matrices, angular velocities, and time interval.
Args:
Cs: The initial attitude matrix.
Cf: The final attitude matrix.
ωs: The initial angular velocity.
ωf: The final angular velocity.
T: The time interval between Cs and Cf.
Returns:
A list of attitude matrices that interpolate between Cs and Cf.
"""
# Interpolate the attitude matrices.
C = smooth_attitude_interpolation(Cs, Cf, ωs, ωf, T)
# Check if the solution is valid.
if not np.allclose(C[0], Cs):
raise ValueError("The initial attitude matrix is not preserved.")
if not np.allclose(C[-1], Cf):
raise ValueError("The final attitude matrix is not preserved.")
return C
Cs = np.array([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) Cf = np.array([[0, 0, 1], [0, 1, 0], [-1, 0, 0]]) ωs = np.array([0, 0, 1]) ωf = np.array([0, 0, -1]) T = 1.0
C = solve_np_c(Cs, Cf, ωs, ωf, T)