Skip to content

RTN Transformations

Rotation matrices between ECI (Earth-Centered Inertial) and RTN (Radial-Tangential-Normal) orbital frames.

Note

For conceptual explanations of the RTN frame, see Relative Motion.

rotation_rtn_to_eci builtin 

rotation_rtn_to_eci(x_eci: Union[ndarray, List]) -> ndarray

Computes the rotation matrix transforming a vector in the radial, along-track, cross-track (RTN) frame to the Earth-Centered Inertial (ECI) frame.

The ECI frame can be any inertial frame centered at the Earth's center, such as GCRF or EME2000.

The RTN frame is defined as follows: - R (Radial): Points from the Earth's center to the satellite's position. - N (Cross-Track): Perpendicular to the orbital plane, defined by the angular momentum vector (cross product of position and velocity). - T (Along-Track): Completes the right-handed coordinate system, lying in the orbital plane and perpendicular to R and N.

Parameters:

Name Type Description Default
x_eci ndarray or list

6D state vector in the ECI frame [x, y, z, vx, vy, vz] (m, m/s), shape (6,)

 required

Returns:

Type Description
ndarray

numpy.ndarray: 3x3 rotation matrix transforming from RTN to ECI frame, shape (3, 3)
Example 
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
import brahe as bh
import numpy as np

# Define satellite state
sma = bh.R_EARTH + 700e3  # Semi-major axis in meters
state = np.array([sma, 0.0, 0.0, 0.0, bh.perigee_velocity(sma, 0.0), 0.0])

# Get rotation matrix
R = bh.rotation_rtn_to_eci(state)
print(f"RTN to ECI rotation matrix:\n{R}")

rotation_eci_to_rtn builtin 

rotation_eci_to_rtn(x_eci: Union[ndarray, List]) -> ndarray

Computes the rotation matrix transforming a vector in the Earth-Centered Inertial (ECI) frame to the radial, along-track, cross-track (RTN) frame.

This is the transpose (inverse) of the RTN-to-ECI rotation matrix.

Parameters:

Name Type Description Default
x_eci ndarray or list

6D state vector in the ECI frame [x, y, z, vx, vy, vz] (m, m/s), shape (6,)

 required

Returns:

Type Description
ndarray

numpy.ndarray: 3x3 rotation matrix transforming from ECI to RTN frame, shape (3, 3)
Example 
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
import brahe as bh
import numpy as np

# Define satellite state
sma = bh.R_EARTH + 700e3  # Semi-major axis in meters
state = np.array([sma, 0.0, 0.0, 0.0, bh.perigee_velocity(sma, 0.0), 0.0])

# Get rotation matrix
R = bh.rotation_eci_to_rtn(state)
print(f"ECI to RTN rotation matrix:\n{R}")

state_rtn_to_eci builtin 

state_rtn_to_eci(x_chief: Union[ndarray, List], x_rel_rtn: Union[ndarray, List]) -> ndarray

Transforms the relative state of a deputy satellite with respect to a chief satellite from the rotating Radial, Along-Track, Cross-Track (RTN) frame to the absolute state of the deputy in the Earth-Centered Inertial (ECI) frame.

Parameters:

Name Type Description Default
x_chief ndarray or list

6D state vector of the chief satellite in the ECI frame [x, y, z, vx, vy, vz] (m, m/s), shape (6,)

 required
x_rel_rtn ndarray or list

6D relative state vector of the deputy with respect to the chief in the RTN frame [ρ_R, ρ_T, ρ_N, ρ̇_R, ρ̇_T, ρ̇_N] (m, m/s), shape (6,)

 required

Returns:

Type Description
ndarray

numpy.ndarray: 6D state vector of the deputy satellite in the ECI frame [x, y, z, vx, vy, vz] (m, m/s), shape (6,)
Example 
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
import brahe as bh
import numpy as np

bh.initialize_eop()

# Define chief state and relative RTN state
oe_chief = np.array([bh.R_EARTH + 700e3, 0.001, 97.8, 15.0, 30.0, 45.0])
x_chief = bh.state_koe_to_eci(oe_chief, bh.AngleFormat.DEGREES)

# Relative state: 1km radial, 0.5km along-track, -0.3km cross-track
x_rel_rtn = np.array([1000.0, 500.0, -300.0, 0.0, 0.0, 0.0])

# Transform to absolute deputy ECI state
x_deputy = bh.state_rtn_to_eci(x_chief, x_rel_rtn)
print(f"Deputy state in ECI: {x_deputy}")

state_eci_to_rtn builtin 

state_eci_to_rtn(x_chief: Union[ndarray, List], x_deputy: Union[ndarray, List]) -> ndarray

Transforms the absolute states of a chief and deputy satellite from the Earth-Centered Inertial (ECI) frame to the relative state of the deputy with respect to the chief in the rotating Radial, Along-Track, Cross-Track (RTN) frame.

Parameters:

Name Type Description Default
x_chief ndarray or list

6D state vector of the chief satellite in the ECI frame [x, y, z, vx, vy, vz] (m, m/s), shape (6,)

 required
x_deputy ndarray or list

6D state vector of the deputy satellite in the ECI frame [x, y, z, vx, vy, vz] (m, m/s), shape (6,)

 required

Returns:

Type Description
ndarray

numpy.ndarray: 6D relative state vector of the deputy with respect to the chief in the RTN frame [ρ_R, ρ_T, ρ_N, ρ̇_R, ρ̇_T, ρ̇_N] (m, m/s), shape (6,)
Example 
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
import brahe as bh
import numpy as np

bh.initialize_eop()

# Define chief and deputy orbital elements
oe_chief = np.array([bh.R_EARTH + 700e3, 0.001, 97.8, 15.0, 30.0, 45.0])
oe_deputy = np.array([bh.R_EARTH + 701e3, 0.0015, 97.85, 15.05, 30.05, 45.05])

# Convert to Cartesian states
x_chief = bh.state_koe_to_eci(oe_chief, bh.AngleFormat.DEGREES)
x_deputy = bh.state_koe_to_eci(oe_deputy, bh.AngleFormat.DEGREES)

# Transform to relative RTN state
x_rel_rtn = bh.state_eci_to_rtn(x_chief, x_deputy)
print(f"Relative state in RTN: {x_rel_rtn}")