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, Sequence], axis: int = -1) -> 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,), or a batch of vectors with the 6 components along `axis` (for example shape (n, 6)).	required
`axis`	`int`	The axis along which the 6 components of a single vector lie; the remaining axes enumerate the batch. For a batch of shape (n, 6) the components lie along the last axis, so the default `-1` applies; a (6, n) column layout uses `axis=0`.	`-1`

Returns:

Type	Description
`ndarray`	numpy.ndarray: 3x3 rotation matrix transforming from RTN to ECI frame, shape (3, 3), or the batch dimensions followed by (3, 3) for batched input.

Example

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, Sequence], axis: int = -1) -> 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,), or a batch of vectors with the 6 components along `axis` (for example shape (n, 6)).	required
`axis`	`int`	The axis along which the 6 components of a single vector lie; the remaining axes enumerate the batch. For a batch of shape (n, 6) the components lie along the last axis, so the default `-1` applies; a (6, n) column layout uses `axis=0`.	`-1`

Returns:

Type	Description
`ndarray`	numpy.ndarray: 3x3 rotation matrix transforming from ECI to RTN frame, shape (3, 3), or the batch dimensions followed by (3, 3) for batched input.

Example

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, Sequence], x_rel_rtn: Union[ndarray, Sequence], axis: int = -1) -> 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,), or a batch of vectors with the 6 components along `axis` (for example shape (n, 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,), or a batch of vectors with the 6 components along `axis` (for example shape (n, 6)).	required
`axis`	`int`	The axis along which the 6 components of a single vector lie; the remaining axes enumerate the batch. For a batch of shape (n, 6) the components lie along the last axis, so the default `-1` applies; a (6, n) column layout uses `axis=0`. A single vector in one argument is broadcast across a batch in the other.	`-1`

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,), or the layout of the batched argument for batched input.

Example

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, Sequence], x_deputy: Union[ndarray, Sequence], axis: int = -1) -> 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,), or a batch of vectors with the 6 components along `axis` (for example shape (n, 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,), or a batch of vectors with the 6 components along `axis` (for example shape (n, 6)).	required
`axis`	`int`	The axis along which the 6 components of a single vector lie; the remaining axes enumerate the batch. For a batch of shape (n, 6) the components lie along the last axis, so the default `-1` applies; a (6, n) column layout uses `axis=0`. A single vector in one argument is broadcast across a batch in the other.	`-1`

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,), or the layout of the batched argument for batched input.

Example

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}")

RTN Transformations¶

rotation_rtn_to_eci builtin ¶

rotation_eci_to_rtn builtin ¶

state_rtn_to_eci builtin ¶

state_eci_to_rtn builtin ¶

rotation_rtn_to_eci `builtin` ¶

rotation_eci_to_rtn `builtin` ¶

state_rtn_to_eci `builtin` ¶

state_eci_to_rtn `builtin` ¶