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Frame Transformations

Brahe provides functions for transforming between different reference frames.

Cartesian Inputs

All frame transformation functions in Brahe assume the input state are in Cartesian coordinates (position and velocity).

Earth-Centered Inertial (ECI) and Earth-Centered Earth-Fixed (ECEF) Transformations

We can transform from ECI to ECEF using state_eci_to_ecef, which properly accounts for Earth's precession, nutation, sidereal rotation, and polar motion.

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import brahe as bh
import numpy as np

# Initialize EOP
bh.initialize_eop()

# Create an epoch
epc = bh.Epoch(2024, 1, 1, 0, 0, 0)

# Initialize a Keplerian state
# Define orbital elements [a, e, i, Ω, ω, M] in meters and degrees
# LEO satellite: 500 km altitude, 97.8° inclination (approx sun-synchronous)
oe_deg = np.array(
    [
        bh.R_EARTH + 500e3,  # Semi-major axis (m)
        0.01,  # Eccentricity
        97.8,  # Inclination (deg)
        15.0,  # Right ascension of ascending node (deg)
        30.0,  # Argument of periapsis (deg)
        45.0,  # Mean anomaly (deg)
    ]
)

# Convert orbital elements to Cartesian state using degrees
x_eci_1 = bh.state_koe_to_eci(oe_deg, bh.AngleFormat.DEGREES)

# Covert ECI cartesian state to ECEF cartesian state
x_ecef = bh.state_eci_to_ecef(epc, x_eci_1)

# Convert ECEF back to ECI to verify consistency
x_eci_2 = bh.state_ecef_to_eci(epc, x_ecef)
print(f"ECI -> ECEF -> ECI rountrip difference: {np.linalg.norm(x_eci_2 - x_eci_1):.3e}")

# Perform same transformation with GCRF/ITRF naming

x_gcrf_1 = x_eci_1
x_itrf = bh.state_gcrf_to_itrf(epc, x_gcrf_1)
x_gcrf_2 = bh.state_itrf_to_gcrf(epc, x_itrf)
print(f"GCRF -> ITRF -> GCRF rountrip difference: {np.linalg.norm(x_gcrf_2 - x_gcrf_1):.3e}")

print(f'ECEF <> ITRF difference: {np.linalg.norm(x_ecef - x_itrf):.3e}')

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use brahe as bh;
use nalgebra as na;

fn main() {
    // Initialize EOP
    bh::initialize_eop().unwrap();

    // Create an epoch
    let epc = bh::Epoch::from_datetime(2024, 1, 1, 0, 0, 0.0, 0.0, bh::TimeSystem::UTC);

    // Initialize a Keplerian state
    // Define orbital elements [a, e, i, Ω, ω, M] in meters and degrees
    // LEO satellite: 500 km altitude, 97.8° inclination (approx sun-synchronous)
    let oe_deg = na::vector![
        bh::R_EARTH + 500e3,  // Semi-major axis (m)
        0.01,                  // Eccentricity
        97.8,                  // Inclination (deg)
        15.0,                  // Right ascension of ascending node (deg)
        30.0,                  // Argument of periapsis (deg)
        45.0,                  // Mean anomaly (deg)
    ];

    // Convert orbital elements to Cartesian state using degrees
    let x_eci_1 = bh::state_koe_to_eci(oe_deg, bh::AngleFormat::Degrees);

    // Convert ECI cartesian state to ECEF cartesian state
    let x_ecef = bh::state_eci_to_ecef(epc, x_eci_1);

    // Convert ECEF back to ECI to verify consistency
    let x_eci_2 = bh::state_ecef_to_eci(epc, x_ecef);
    let eci_roundtrip_diff = (x_eci_2 - x_eci_1).norm();
    println!("ECI -> ECEF -> ECI roundtrip difference: {:.3e}", eci_roundtrip_diff);

    // Perform same transformation with GCRF/ITRF naming
    let x_gcrf_1 = x_eci_1;
    let x_itrf = bh::state_gcrf_to_itrf(epc, x_gcrf_1);
    let x_gcrf_2 = bh::state_itrf_to_gcrf(epc, x_itrf);
    let gcrf_roundtrip_diff = (x_gcrf_2 - x_gcrf_1).norm();
    println!("GCRF -> ITRF -> GCRF roundtrip difference: {:.3e}", gcrf_roundtrip_diff);

    let ecef_itrf_diff = (x_ecef - x_itrf).norm();
    println!("ECEF <> ITRF difference: {:.3e}", ecef_itrf_diff);
}
Output 
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ECI -> ECEF -> ECI rountrip difference: 9.331e-10
GCRF -> ITRF -> GCRF rountrip difference: 9.331e-10
ECEF <> ITRF difference: 0.000e+00

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ECI -> ECEF -> ECI roundtrip difference: 9.331e-10
GCRF -> ITRF -> GCRF roundtrip difference: 9.331e-10
ECEF <> ITRF difference: 0.000e0