Covariance and Sensitivity¶

The NumericalOrbitPropagator can propagate additional quantities alongside the orbital state, enabling covariance propagation and sensitivity analysis. This is essential for uncertainty quantification, orbit determination, and mission analysis.

Full Example¶

Here is a complete example demonstrating STM, covariance, and sensitivity propagation together:

PythonRust

import numpy as np
import brahe as bh

# Initialize EOP and space weather data (required for NRLMSISE-00 drag model)
bh.initialize_eop()
bh.initialize_sw()

# Create initial epoch and state (LEO satellite)
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 500e3, 0.01, 45.0, 15.0, 30.0, 45.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Define spacecraft parameters: [mass, drag_area, Cd, srp_area, Cr]
params = np.array([500.0, 2.0, 2.2, 2.0, 1.3])

# Create propagation config enabling STM and sensitivity with history storage
prop_config = (
    bh.NumericalPropagationConfig.default()
    .with_stm()
    .with_stm_history()
    .with_sensitivity()
    .with_sensitivity_history()
)

# Define initial covariance (diagonal)
# Position uncertainty: 10 m (variance = 100 m²)
# Velocity uncertainty: 0.01 m/s (variance = 0.0001 m²/s²)
P0 = np.diag([100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001])

# Create propagator with full force model
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.default(),
    params=params,
    initial_covariance=P0,
)

print("=== Variational Propagation Overview ===\n")
print("Initial State:")
print(f"  Semi-major axis: {oe[0] / 1000:.1f} km")
print(f"  Position std: {np.sqrt(P0[0, 0]):.1f} m")
print(f"  Velocity std: {np.sqrt(P0[3, 3]) * 1000:.2f} mm/s")

# Propagate for one orbital period
orbital_period = bh.orbital_period(oe[0])
prop.propagate_to(epoch + orbital_period)

# === STM Access ===
print("\n--- State Transition Matrix (STM) ---")
stm = prop.stm()
print(f"STM shape: {stm.shape}")
print(
    f"STM determinant: {np.linalg.det(stm):.6f} (should be ~1 for conservative forces)"
)

# STM at intermediate time (half orbit)
stm_half = prop.stm_at(epoch + orbital_period / 2)
print(f"STM at t/2 available: {stm_half is not None}")

# === Covariance Propagation ===
print("\n--- Covariance Propagation ---")

# Manual propagation: P(t) = STM @ P0 @ STM^T
P_manual = stm @ P0 @ stm.T

# Using built-in covariance retrieval
P_gcrf = prop.covariance_gcrf(epoch + orbital_period)
P_rtn = prop.covariance_rtn(epoch + orbital_period)

# Extract position uncertainties
pos_std_gcrf = np.sqrt(np.diag(P_gcrf[:3, :3]))
pos_std_rtn = np.sqrt(np.diag(P_rtn[:3, :3]))

print("Position std (GCRF frame):")
print(
    f"  X: {pos_std_gcrf[0]:.1f} m, Y: {pos_std_gcrf[1]:.1f} m, Z: {pos_std_gcrf[2]:.1f} m"
)
print("Position std (RTN frame):")
print(
    f"  R: {pos_std_rtn[0]:.1f} m, T: {pos_std_rtn[1]:.1f} m, N: {pos_std_rtn[2]:.1f} m"
)

# === Sensitivity Analysis ===
print("\n--- Parameter Sensitivity ---")
sens = prop.sensitivity()
print(f"Sensitivity matrix shape: {sens.shape}")

# Position sensitivity magnitude to each parameter
param_names = ["mass", "drag_area", "Cd", "srp_area", "Cr"]
print("\nPosition sensitivity to 1% parameter uncertainty:")
for i, name in enumerate(param_names):
    pos_sens_mag = np.linalg.norm(sens[:3, i])
    param_uncertainty = params[i] * 0.01  # 1% uncertainty
    pos_error = pos_sens_mag * param_uncertainty
    print(f"  {name:10s}: {pos_error:.2f} m")

# === Summary ===
print("\n--- Summary ---")
total_pos_std_initial = np.sqrt(np.trace(P0[:3, :3]))
total_pos_std_final = np.sqrt(np.trace(P_gcrf[:3, :3]))
print(
    f"Total position uncertainty: {total_pos_std_initial:.1f} m -> {total_pos_std_final:.1f} m"
)
print(f"Uncertainty growth factor: {total_pos_std_final / total_pos_std_initial:.1f}x")

# Validate outputs
assert stm.shape == (6, 6)
assert sens.shape == (6, 5)
assert P_gcrf.shape == (6, 6)
assert P_rtn.shape == (6, 6)
assert total_pos_std_final >= total_pos_std_initial

print("\nExample validated successfully!")

use brahe as bh;
use bh::traits::{DOrbitCovarianceProvider, DStatePropagator};
use nalgebra as na;
use std::f64::consts::PI;

fn main() {
    // Initialize EOP and space weather data (required for NRLMSISE-00 drag model)
    bh::initialize_eop().unwrap();
    bh::initialize_sw().unwrap();

    // Create initial epoch and state (LEO satellite)
    let epoch = bh::Epoch::from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh::TimeSystem::UTC);
    let oe = na::SVector::<f64, 6>::new(bh::R_EARTH + 500e3, 0.01, 45.0, 0.0, 0.0, 0.0);
    let state = bh::state_koe_to_eci(oe, bh::AngleFormat::Degrees);

    // Define spacecraft parameters: [mass, drag_area, Cd, srp_area, Cr]
    let params = na::DVector::from_vec(vec![500.0, 2.0, 2.2, 2.0, 1.3]);

    // Create propagation config enabling STM and sensitivity with history storage
    let mut prop_config = bh::NumericalPropagationConfig::default();
    prop_config.variational.enable_stm = true;
    prop_config.variational.store_stm_history = true;
    prop_config.variational.enable_sensitivity = true;
    prop_config.variational.store_sensitivity_history = true;

    // Define initial covariance (diagonal)
    // Position uncertainty: 10 m (variance = 100 m²)
    // Velocity uncertainty: 0.01 m/s (variance = 0.0001 m²/s²)
    let p0: na::DMatrix<f64> = na::DMatrix::from_diagonal(&na::DVector::from_vec(vec![
        100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001,
    ]));

    // Create propagator with full force model
    // Arguments: epoch, state, config, force_config, params, additional_dynamics, control_input, initial_covariance
    let mut prop = bh::DNumericalOrbitPropagator::new(
        epoch,
        na::DVector::from_column_slice(state.as_slice()),
        prop_config,
        bh::ForceModelConfig::default(),
        Some(params.clone()),
        None,  // additional_dynamics
        None,  // control_input
        Some(p0.clone()),  // initial_covariance
    )
    .unwrap();

    println!("=== Variational Propagation Overview ===\n");
    println!("Initial State:");
    println!("  Semi-major axis: {:.1} km", oe[0] / 1000.0);
    println!("  Position std: {:.1} m", p0[(0, 0)].sqrt());
    println!("  Velocity std: {:.2} mm/s", p0[(3, 3)].sqrt() * 1000.0);

    // Propagate for one orbital period
    let orbital_period = 2.0 * PI * (oe[0].powi(3) / bh::GM_EARTH).sqrt();
    prop.propagate_to(epoch + orbital_period);

    // === STM Access ===
    println!("\n--- State Transition Matrix (STM) ---");
    let stm = prop.stm().expect("STM should be available");
    println!("STM shape: ({}, {})", stm.nrows(), stm.ncols());
    println!(
        "STM determinant: {:.6} (should be ~1 for conservative forces)",
        stm.determinant()
    );

    // STM at intermediate time (half orbit)
    let stm_half = prop.stm_at(epoch + orbital_period / 2.0);
    println!("STM at t/2 available: {}", stm_half.is_some());

    // === Covariance Propagation ===
    println!("\n--- Covariance Propagation ---");

    // Manual propagation: P(t) = STM @ P0 @ STM^T
    let _p_manual = stm * &p0 * stm.transpose();

    // Using built-in covariance retrieval
    let p_gcrf = prop.covariance_gcrf(epoch + orbital_period).unwrap();
    let p_rtn = prop.covariance_rtn(epoch + orbital_period).unwrap();

    // Extract position uncertainties
    println!("Position std (GCRF frame):");
    println!(
        "  X: {:.1} m, Y: {:.1} m, Z: {:.1} m",
        p_gcrf[(0, 0)].sqrt(),
        p_gcrf[(1, 1)].sqrt(),
        p_gcrf[(2, 2)].sqrt()
    );
    println!("Position std (RTN frame):");
    println!(
        "  R: {:.1} m, T: {:.1} m, N: {:.1} m",
        p_rtn[(0, 0)].sqrt(),
        p_rtn[(1, 1)].sqrt(),
        p_rtn[(2, 2)].sqrt()
    );

    // === Sensitivity Analysis ===
    println!("\n--- Parameter Sensitivity ---");
    let sens = prop.sensitivity().expect("Sensitivity should be available");
    println!("Sensitivity matrix shape: ({}, {})", sens.nrows(), sens.ncols());

    // Position sensitivity magnitude to each parameter
    let param_names = ["mass", "drag_area", "Cd", "srp_area", "Cr"];
    println!("\nPosition sensitivity to 1% parameter uncertainty:");
    for (i, name) in param_names.iter().enumerate() {
        let pos_sens = na::Vector3::new(sens[(0, i)], sens[(1, i)], sens[(2, i)]);
        let pos_sens_mag = pos_sens.norm();
        let param_uncertainty = params[i] * 0.01; // 1% uncertainty
        let pos_error = pos_sens_mag * param_uncertainty;
        println!("  {:10}: {:.2} m", name, pos_error);
    }

    // === Summary ===
    println!("\n--- Summary ---");
    let total_pos_std_initial = (p0[(0, 0)] + p0[(1, 1)] + p0[(2, 2)]).sqrt();
    let total_pos_std_final = (p_gcrf[(0, 0)] + p_gcrf[(1, 1)] + p_gcrf[(2, 2)]).sqrt();
    println!(
        "Total position uncertainty: {:.1} m -> {:.1} m",
        total_pos_std_initial, total_pos_std_final
    );
    println!(
        "Uncertainty growth factor: {:.1}x",
        total_pos_std_final / total_pos_std_initial
    );

    // Validate outputs
    assert_eq!(stm.nrows(), 6);
    assert_eq!(stm.ncols(), 6);
    assert_eq!(sens.nrows(), 6);
    assert_eq!(sens.ncols(), 5);
    assert_eq!(p_gcrf.nrows(), 6);
    assert_eq!(p_rtn.nrows(), 6);
    assert!(total_pos_std_final >= total_pos_std_initial);

    println!("\nExample validated successfully!");
}

Output

PythonRust

=== Variational Propagation Overview ===

Initial State:
  Semi-major axis: 6878.1 km
  Position std: 10.0 m
  Velocity std: 10.00 mm/s

--- State Transition Matrix (STM) ---
STM shape: (6, 6)
STM determinant: 0.999999 (should be ~1 for conservative forces)
STM at t/2 available: True

--- Covariance Propagation ---
Position std (GCRF frame):
  X: 254.8 m, Y: 26.1 m, Z: 41.4 m
Position std (RTN frame):
  R: 8.7 m, T: 259.1 m, N: 10.0 m

--- Parameter Sensitivity ---
Sensitivity matrix shape: (6, 5)

Position sensitivity to 1% parameter uncertainty:
  mass      : 0.04 m
  drag_area : 0.04 m
  Cd        : 0.04 m
  srp_area  : 0.00 m
  Cr        : 0.00 m

--- Summary ---
Total position uncertainty: 17.3 m -> 259.4 m
Uncertainty growth factor: 15.0x

Example validated successfully!

=== Variational Propagation Overview ===

Initial State:
  Semi-major axis: 6878.1 km
  Position std: 10.0 m
  Velocity std: 10.00 mm/s

--- State Transition Matrix (STM) ---
STM shape: (6, 6)
STM determinant: 0.999999 (should be ~1 for conservative forces)
STM at t/2 available: true

--- Covariance Propagation ---
Position std (GCRF frame):
  X: 12.3 m, Y: 184.2 m, Z: 184.4 m
Position std (RTN frame):
  R: 10.0 m, T: 260.5 m, N: 10.0 m

--- Parameter Sensitivity ---
Sensitivity matrix shape: (6, 5)

Position sensitivity to 1% parameter uncertainty:
  mass      : 0.08 m
  drag_area : 0.08 m
  Cd        : 0.08 m
  srp_area  : 0.00 m
  Cr        : 0.00 m

--- Summary ---
Total position uncertainty: 17.3 m -> 260.9 m
Uncertainty growth factor: 15.1x

Example validated successfully!

Architecture Overview¶

Configuration Hierarchy¶

Variational propagation is configured through the VariationalConfig within NumericalPropagationConfig:

NumericalPropagationConfig
├── method: IntegrationMethod
├── integrator: IntegratorConfig
└── variational: VariationalConfig
    ├── enable_stm: bool
    ├── enable_sensitivity: bool
    ├── store_stm_history: bool
    ├── store_sensitivity_history: bool
    ├── jacobian_method: DifferenceMethod
    └── sensitivity_method: DifferenceMethod

Auto-Enable Behavior¶

Providing initial_covariance when creating the propagator automatically enables STM propagation, even without explicitly setting enable_stm = true.

State Transition Matrices (STM)¶

The State Transition Matrix (STM) is a foundational tool for linear uncertainty propagation. It describes how small perturbations in the initial state map to perturbations at a later time:

\[\delta \mathbf{x}(t) = \Phi(t, t_0) \delta \mathbf{x}(t_0)\]

where \(\Phi(t, t_0)\) is the 6x6 STM from epoch \(t_0\) to time \(t\).

STM Structure¶

For orbital mechanics with state \(\mathbf{x} = [x, y, z, v_x, v_y, v_z]^T\), the 6x6 STM has structure:

\[\Phi = \begin{bmatrix} \frac{\partial \mathbf{r}}{\partial \mathbf{r}_0} & \frac{\partial \mathbf{r}}{\partial \mathbf{v}_0} \\ \frac{\partial \mathbf{v}}{\partial \mathbf{r}_0} & \frac{\partial \mathbf{v}}{\partial \mathbf{v}_0} \end{bmatrix}\]

Submatrix	Location	Physical Meaning
\(\frac{\partial \mathbf{r}}{\partial \mathbf{r}_0}\)	Upper left	Position sensitivity to initial position
\(\frac{\partial \mathbf{r}}{\partial \mathbf{v}_0}\)	Upper right	Position sensitivity to initial velocity
\(\frac{\partial \mathbf{v}}{\partial \mathbf{r}_0}\)	Lower left	Velocity sensitivity to initial position
\(\frac{\partial \mathbf{v}}{\partial \mathbf{v}_0}\)	Lower right	Velocity sensitivity to initial velocity

STM Properties¶

The STM has several important mathematical properties:

Identity at initial time: \(\Phi(t_0, t_0) = I\)
Composition: STMs can be composed to span longer intervals: \(\Phi(t_2, t_0) = \Phi(t_2, t_1) \Phi(t_1, t_0)\)
Determinant preservation: For Hamiltonian systems (conservative forces only), \(\det(\Phi) = 1\)

Enabling STM Propagation¶

PythonRust

import numpy as np
import brahe as bh

# Initialize EOP data
bh.initialize_eop()

# Create initial epoch and state
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 500e3, 0.01, 45.0, 15.0, 30.0, 45.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Method 1: Enable STM via builder pattern
prop_config = bh.NumericalPropagationConfig.default().with_stm().with_stm_history()

# Create propagator with two-body gravity
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.two_body(),
    None,
)

print("=== STM Propagation Example ===\n")

# Propagate for one orbital period
orbital_period = bh.orbital_period(oe[0])
prop.propagate_to(epoch + orbital_period)

# Access STM at final time
stm = prop.stm()
print(f"Final STM shape: {stm.shape}")
print(f"STM determinant: {np.linalg.det(stm):.6f}")

# STM at initial time should be identity
stm_initial = prop.stm_at(epoch)
print("\nSTM at t=0 (should be identity):")
print(f"  Max off-diagonal: {np.max(np.abs(stm_initial - np.eye(6))):.2e}")

# STM at intermediate time
half_period = epoch + orbital_period / 2
stm_half = prop.stm_at(half_period)
print("\nSTM at t=T/2:")
print(f"  Determinant: {np.linalg.det(stm_half):.6f}")

# STM composition property: Phi(t2,t0) = Phi(t2,t1) * Phi(t1,t0)
# For verification, we check that the STM is invertible
stm_inv = np.linalg.inv(stm)
identity_check = stm @ stm_inv
print("\nSTM * STM^-1 (should be identity):")
print(f"  Max deviation from I: {np.max(np.abs(identity_check - np.eye(6))):.2e}")

# STM structure interpretation
print("\n=== STM Structure ===")
print("Upper-left 3x3: Position sensitivity to initial position")
print("Upper-right 3x3: Position sensitivity to initial velocity")
print("Lower-left 3x3: Velocity sensitivity to initial position")
print("Lower-right 3x3: Velocity sensitivity to initial velocity")

# Show magnitude of each block
pos_pos = np.linalg.norm(stm[:3, :3])
pos_vel = np.linalg.norm(stm[:3, 3:])
vel_pos = np.linalg.norm(stm[3:, :3])
vel_vel = np.linalg.norm(stm[3:, 3:])

print("\nBlock Frobenius norms after one orbit:")
print(f"  dr/dr0: {pos_pos:.2f}")
print(f"  dr/dv0: {pos_vel:.2f}")
print(f"  dv/dr0: {vel_pos:.6f}")
print(f"  dv/dv0: {vel_vel:.2f}")

# Validate
assert stm.shape == (6, 6)
assert np.abs(np.linalg.det(stm) - 1.0) < 1e-6  # Hamiltonian system preserves volume
assert stm_initial is not None
assert stm_half is not None

print("\nExample validated successfully!")

use brahe as bh;
use bh::traits::DStatePropagator;
use nalgebra as na;
use std::f64::consts::PI;

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

    // Create initial epoch and state
    let epoch = bh::Epoch::from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh::TimeSystem::UTC);
    let oe = na::SVector::<f64, 6>::new(bh::R_EARTH + 500e3, 0.01, 45.0, 0.0, 0.0, 0.0);
    let state = bh::state_koe_to_eci(oe, bh::AngleFormat::Degrees);

    // Enable STM via config
    let mut prop_config = bh::NumericalPropagationConfig::default();
    prop_config.variational.enable_stm = true;
    prop_config.variational.store_stm_history = true;

    // Create propagator with two-body gravity
    let mut prop = bh::DNumericalOrbitPropagator::new(
        epoch,
        na::DVector::from_column_slice(state.as_slice()),
        prop_config,
        bh::ForceModelConfig::two_body_gravity(),
        None,
        None,
        None,
        None,
    )
    .unwrap();

    println!("=== STM Propagation Example ===\n");

    // Propagate for one orbital period
    let orbital_period = 2.0 * PI * (oe[0].powi(3) / bh::GM_EARTH).sqrt();
    prop.propagate_to(epoch + orbital_period);

    // Access STM at final time
    let stm = prop.stm().expect("STM should be available");
    println!("Final STM shape: ({}, {})", stm.nrows(), stm.ncols());
    println!("STM determinant: {:.6}", stm.determinant());

    // STM at initial time should be identity
    let stm_initial = prop.stm_at(epoch).expect("STM at t=0 should be available");
    let identity = na::DMatrix::identity(6, 6);
    let max_diff = (&stm_initial - &identity).abs().max();
    println!("\nSTM at t=0 (should be identity):");
    println!("  Max off-diagonal: {:.2e}", max_diff);

    // STM at intermediate time
    let half_period = epoch + orbital_period / 2.0;
    let stm_half = prop.stm_at(half_period).expect("STM at t=T/2 should be available");
    println!("\nSTM at t=T/2:");
    println!("  Determinant: {:.6}", stm_half.determinant());

    // STM composition property verification
    let stm_inv = stm.clone().try_inverse().expect("STM should be invertible");
    let identity_check = stm * &stm_inv;
    let max_deviation = (&identity_check - &identity).abs().max();
    println!("\nSTM * STM^-1 (should be identity):");
    println!("  Max deviation from I: {:.2e}", max_deviation);

    // STM structure interpretation
    println!("\n=== STM Structure ===");
    println!("Upper-left 3x3: Position sensitivity to initial position");
    println!("Upper-right 3x3: Position sensitivity to initial velocity");
    println!("Lower-left 3x3: Velocity sensitivity to initial position");
    println!("Lower-right 3x3: Velocity sensitivity to initial velocity");

    // Show magnitude of each block
    let stm = prop.stm().unwrap();
    let pos_pos = stm.view((0, 0), (3, 3)).norm();
    let pos_vel = stm.view((0, 3), (3, 3)).norm();
    let vel_pos = stm.view((3, 0), (3, 3)).norm();
    let vel_vel = stm.view((3, 3), (3, 3)).norm();

    println!("\nBlock Frobenius norms after one orbit:");
    println!("  dr/dr0: {:.2}", pos_pos);
    println!("  dr/dv0: {:.2}", pos_vel);
    println!("  dv/dr0: {:.6}", vel_pos);
    println!("  dv/dv0: {:.2}", vel_vel);

    // Validate
    assert_eq!(stm.nrows(), 6);
    assert_eq!(stm.ncols(), 6);
    assert!((stm.determinant() - 1.0).abs() < 1e-6); // Hamiltonian system preserves volume

    println!("\nExample validated successfully!");
}

Output

PythonRust

=== STM Propagation Example ===

Final STM shape: (6, 6)
STM determinant: 1.000000

STM at t=0 (should be identity):
  Max off-diagonal: 0.00e+00

STM at t=T/2:
  Determinant: 0.999999

STM * STM^-1 (should be identity):
  Max deviation from I: 5.45e-12

=== STM Structure ===
Upper-left 3x3: Position sensitivity to initial position
Upper-right 3x3: Position sensitivity to initial velocity
Lower-left 3x3: Velocity sensitivity to initial position
Lower-right 3x3: Velocity sensitivity to initial velocity

Block Frobenius norms after one orbit:
  dr/dr0: 19.32
  dr/dv0: 17271.76
  dv/dr0: 0.021459
  dv/dv0: 19.34

Example validated successfully!

=== STM Propagation Example ===

Final STM shape: (6, 6)
STM determinant: 1.000000

STM at t=0 (should be identity):
  Max off-diagonal: 0.00e0

STM at t=T/2:
  Determinant: 0.999999

STM * STM^-1 (should be identity):
  Max deviation from I: 2.38e-12

=== STM Structure ===
Upper-left 3x3: Position sensitivity to initial position
Upper-right 3x3: Position sensitivity to initial velocity
Lower-left 3x3: Velocity sensitivity to initial position
Lower-right 3x3: Velocity sensitivity to initial velocity

Block Frobenius norms after one orbit:
  dr/dr0: 19.50
  dr/dv0: 17375.04
  dv/dr0: 0.021718
  dv/dv0: 19.50

Example validated successfully!

Covariance Propagation¶

The primary application of the STM is propagating uncertainty. Given an initial covariance \(P_0\), the propagated covariance is:

\[P(t) = \Phi(t, t_0) P_0 \Phi(t, t_0)^T\]

Creating a Propagator with Initial Covariance¶

PythonRust

import numpy as np
import brahe as bh

# Initialize EOP data
bh.initialize_eop()

# Create initial epoch and state
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 500e3, 0.01, 45.0, 15.0, 30.0, 45.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Create propagation config with STM enabled
prop_config = bh.NumericalPropagationConfig.default().with_stm()

# Create propagator (two-body for clean demonstration)
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.two_body(),
    None,
)

# Define initial covariance (diagonal)
# Position uncertainty: 10 m in each axis
# Velocity uncertainty: 0.01 m/s in each axis
P0 = np.diag([100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001])

print("Initial Covariance (diagonal, sqrt):")
print(f"  Position std: {np.sqrt(P0[0, 0]):.1f} m")
print(f"  Velocity std: {np.sqrt(P0[3, 3]) * 1000:.2f} mm/s")

# Propagate for one orbital period
orbital_period = bh.orbital_period(oe[0])
prop.propagate_to(epoch + orbital_period)

# Get the State Transition Matrix
stm = prop.stm()
print(f"\nSTM shape: {stm.shape}")

# Propagate covariance: P(t) = Phi @ P0 @ Phi^T
P = stm @ P0 @ stm.T

# Extract position and velocity uncertainties
pos_cov = P[:3, :3]
vel_cov = P[3:, 3:]

print("\nPropagated Covariance after one orbit:")
print(
    f"  Position std (x,y,z): ({np.sqrt(pos_cov[0, 0]):.1f}, {np.sqrt(pos_cov[1, 1]):.1f}, {np.sqrt(pos_cov[2, 2]):.1f}) m"
)
print(
    f"  Velocity std (x,y,z): ({np.sqrt(vel_cov[0, 0]) * 1000:.2f}, {np.sqrt(vel_cov[1, 1]) * 1000:.2f}, {np.sqrt(vel_cov[2, 2]) * 1000:.2f}) mm/s"
)

# Compute position uncertainty magnitude
pos_uncertainty_initial = np.sqrt(np.trace(P0[:3, :3]))
pos_uncertainty_final = np.sqrt(np.trace(pos_cov))

print("\nTotal position uncertainty:")
print(f"  Initial: {pos_uncertainty_initial:.1f} m")
print(f"  Final:   {pos_uncertainty_final:.1f} m")
print(f"  Growth:  {pos_uncertainty_final / pos_uncertainty_initial:.1f}x")

# Validate that covariance was propagated
assert stm is not None
assert stm.shape == (6, 6)
assert pos_uncertainty_final >= pos_uncertainty_initial  # Uncertainty grows

print("\nExample validated successfully!")

use brahe as bh;
use bh::traits::DStatePropagator;
use nalgebra as na;
use std::f64::consts::PI;

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

    // Create initial epoch and state
    let epoch = bh::Epoch::from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh::TimeSystem::UTC);
    let oe = na::SVector::<f64, 6>::new(bh::R_EARTH + 500e3, 0.01, 45.0, 0.0, 0.0, 0.0);
    let state = bh::state_koe_to_eci(oe, bh::AngleFormat::Degrees);

    // Create propagation config with STM enabled
    let mut prop_config = bh::NumericalPropagationConfig::default();
    prop_config.variational.enable_stm = true;

    // Create propagator (two-body for clean demonstration)
    let mut prop = bh::DNumericalOrbitPropagator::new(
        epoch,
        na::DVector::from_column_slice(state.as_slice()),
        prop_config,
        bh::ForceModelConfig::two_body_gravity(),
        None,
        None,
        None,
        None,
    )
    .unwrap();

    // Define initial covariance (diagonal)
    // Position uncertainty: 10 m in each axis (100 m² variance)
    // Velocity uncertainty: 0.01 m/s in each axis (0.0001 m²/s² variance)
    let p0: na::DMatrix<f64> = na::DMatrix::from_diagonal(&na::DVector::from_vec(vec![
        100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001,
    ]));

    println!("Initial Covariance (diagonal, sqrt):");
    println!("  Position std: {:.1} m", p0[(0, 0)].sqrt());
    println!("  Velocity std: {:.2} mm/s", p0[(3, 3)].sqrt() * 1000.0);

    // Propagate for one orbital period
    let orbital_period = 2.0 * PI * (oe[0].powi(3) / bh::GM_EARTH).sqrt();
    prop.propagate_to(epoch + orbital_period);

    // Get the State Transition Matrix
    let stm = prop.stm().expect("STM should be available").clone();
    println!("\nSTM shape: ({}, {})", stm.nrows(), stm.ncols());

    // Propagate covariance: P(t) = Phi @ P0 @ Phi^T
    let p = &stm * &p0 * stm.transpose();

    // Extract position and velocity uncertainties
    let pos_std_x = p[(0, 0)].sqrt();
    let pos_std_y = p[(1, 1)].sqrt();
    let pos_std_z = p[(2, 2)].sqrt();
    let vel_std_x = p[(3, 3)].sqrt() * 1000.0;
    let vel_std_y = p[(4, 4)].sqrt() * 1000.0;
    let vel_std_z = p[(5, 5)].sqrt() * 1000.0;

    println!("\nPropagated Covariance after one orbit:");
    println!(
        "  Position std (x,y,z): ({:.1}, {:.1}, {:.1}) m",
        pos_std_x, pos_std_y, pos_std_z
    );
    println!(
        "  Velocity std (x,y,z): ({:.2}, {:.2}, {:.2}) mm/s",
        vel_std_x, vel_std_y, vel_std_z
    );

    // Compute position uncertainty magnitude
    let pos_uncertainty_initial = (p0[(0, 0)] + p0[(1, 1)] + p0[(2, 2)]).sqrt();
    let pos_uncertainty_final = (p[(0, 0)] + p[(1, 1)] + p[(2, 2)]).sqrt();

    println!("\nTotal position uncertainty:");
    println!("  Initial: {:.1} m", pos_uncertainty_initial);
    println!("  Final:   {:.1} m", pos_uncertainty_final);
    println!(
        "  Growth:  {:.1}x",
        pos_uncertainty_final / pos_uncertainty_initial
    );

    // Validate that covariance was propagated
    assert_eq!(stm.nrows(), 6);
    assert_eq!(stm.ncols(), 6);
    assert!(pos_uncertainty_final >= pos_uncertainty_initial); // Uncertainty grows

    println!("\nExample validated successfully!");
}

Output

PythonRust

Initial Covariance (diagonal, sqrt):
  Position std: 10.0 m
  Velocity std: 10.00 mm/s

STM shape: (6, 6)

Propagated Covariance after one orbit:
  Position std (x,y,z): (254.3, 27.0, 41.9) m
  Velocity std (x,y,z): (12.96, 208.90, 199.07) mm/s

Total position uncertainty:
  Initial: 17.3 m
  Final:   259.2 m
  Growth:  15.0x

Example validated successfully!

Initial Covariance (diagonal, sqrt):
  Position std: 10.0 m
  Velocity std: 10.00 mm/s

STM shape: (6, 6)

Propagated Covariance after one orbit:
  Position std (x,y,z): (10.0, 184.6, 184.6) m
  Velocity std (x,y,z): (291.55, 10.00, 10.00) mm/s

Total position uncertainty:
  Initial: 17.3 m
  Final:   261.2 m
  Growth:  15.1x

Example validated successfully!

Covariance in RTN Frame¶

The RTN (Radial-Tangential-Normal) frame provides physical insight into how uncertainty evolves relative to the orbit.

PythonRust

import numpy as np
import brahe as bh

# Initialize EOP data
bh.initialize_eop()

# Create initial epoch and state
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 500e3, 0.01, 45.0, 15.0, 30.0, 45.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Enable STM for covariance propagation
prop_config = bh.NumericalPropagationConfig.default().with_stm().with_stm_history()

# Define initial covariance in ECI frame
# Position uncertainty: 10 m in each axis
# Velocity uncertainty: 0.01 m/s in each axis
P0 = np.diag([100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001])

# Create propagator with initial covariance
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.two_body(),
    None,
    initial_covariance=P0,
)

print("=== Covariance in RTN Frame ===\n")
print("Initial position std (ECI): 10.0 m in each axis")

# Propagate for one orbital period
orbital_period = bh.orbital_period(oe[0])
prop.propagate_to(epoch + orbital_period)

# Get covariance in different frames
target = epoch + orbital_period
P_gcrf = prop.covariance_gcrf(target)
P_rtn = prop.covariance_rtn(target)

# Extract position covariances (3x3 upper-left block)
pos_cov_gcrf = P_gcrf[:3, :3]
pos_cov_rtn = P_rtn[:3, :3]

print("\n--- GCRF Frame Results ---")
print("Position std (X, Y, Z):")
print(f"  X: {np.sqrt(pos_cov_gcrf[0, 0]):.1f} m")
print(f"  Y: {np.sqrt(pos_cov_gcrf[1, 1]):.1f} m")
print(f"  Z: {np.sqrt(pos_cov_gcrf[2, 2]):.1f} m")

print("\n--- RTN Frame Results ---")
print("Position std (R, T, N):")
print(f"  Radial (R):     {np.sqrt(pos_cov_rtn[0, 0]):.1f} m  <- Altitude uncertainty")
print(f"  Tangential (T): {np.sqrt(pos_cov_rtn[1, 1]):.1f} m  <- Along-track timing")
print(f"  Normal (N):     {np.sqrt(pos_cov_rtn[2, 2]):.1f} m  <- Cross-track offset")

# Physical interpretation
print("\n--- Physical Interpretation ---")
print("RTN frame aligns with the orbit:")
print("  R (Radial): Points from Earth center to satellite")
print("  T (Tangential): Points along velocity direction")
print("  N (Normal): Completes right-hand system (cross-track)")
print()
print("Key insight: Along-track (T) uncertainty grows fastest because")
print("velocity uncertainty causes timing errors that accumulate.")
print(
    f"After one orbit: T/R ratio = {np.sqrt(pos_cov_rtn[1, 1]) / np.sqrt(pos_cov_rtn[0, 0]):.1f}x"
)

# Show correlation structure
print("\n--- Position Correlation Matrix (RTN) ---")
pos_std_rtn = np.sqrt(np.diag(pos_cov_rtn))
corr_rtn = pos_cov_rtn / np.outer(pos_std_rtn, pos_std_rtn)
print("       R      T      N")
for i, name in enumerate(["R", "T", "N"]):
    print(
        f"  {name}  {corr_rtn[i, 0]:6.3f} {corr_rtn[i, 1]:6.3f} {corr_rtn[i, 2]:6.3f}"
    )

# Validate
assert P_gcrf.shape == (6, 6)
assert P_rtn.shape == (6, 6)
assert np.sqrt(pos_cov_rtn[1, 1]) > np.sqrt(pos_cov_rtn[0, 0])  # T > R

print("\nExample validated successfully!")

use brahe as bh;
use bh::traits::{DOrbitCovarianceProvider, DStatePropagator};
use nalgebra as na;
use std::f64::consts::PI;

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

    // Create initial epoch and state
    let epoch = bh::Epoch::from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh::TimeSystem::UTC);
    let oe = na::SVector::<f64, 6>::new(bh::R_EARTH + 500e3, 0.01, 45.0, 0.0, 0.0, 0.0);
    let state = bh::state_koe_to_eci(oe, bh::AngleFormat::Degrees);

    // Enable STM for covariance propagation
    let mut prop_config = bh::NumericalPropagationConfig::default();
    prop_config.variational.enable_stm = true;
    prop_config.variational.store_stm_history = true;

    // Define initial covariance in ECI frame
    // Position uncertainty: 10 m in each axis
    // Velocity uncertainty: 0.01 m/s in each axis
    let p0: na::DMatrix<f64> = na::DMatrix::from_diagonal(&na::DVector::from_vec(vec![
        100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001,
    ]));

    // Create propagator with initial covariance
    let mut prop = bh::DNumericalOrbitPropagator::new(
        epoch,
        na::DVector::from_column_slice(state.as_slice()),
        prop_config,
        bh::ForceModelConfig::two_body_gravity(),
        None,
        None,
        None,
        Some(p0),
    )
    .unwrap();

    println!("=== Covariance in RTN Frame ===\n");
    println!("Initial position std (ECI): 10.0 m in each axis");

    // Propagate for one orbital period
    let orbital_period = 2.0 * PI * (oe[0].powi(3) / bh::GM_EARTH).sqrt();
    prop.propagate_to(epoch + orbital_period);

    // Get covariance in different frames
    let target = epoch + orbital_period;
    let p_gcrf = prop.covariance_gcrf(target).unwrap();
    let p_rtn = prop.covariance_rtn(target).unwrap();

    println!("\n--- GCRF Frame Results ---");
    println!("Position std (X, Y, Z):");
    println!("  X: {:.1} m", p_gcrf[(0, 0)].sqrt());
    println!("  Y: {:.1} m", p_gcrf[(1, 1)].sqrt());
    println!("  Z: {:.1} m", p_gcrf[(2, 2)].sqrt());

    println!("\n--- RTN Frame Results ---");
    println!("Position std (R, T, N):");
    println!(
        "  Radial (R):     {:.1} m  <- Altitude uncertainty",
        p_rtn[(0, 0)].sqrt()
    );
    println!(
        "  Tangential (T): {:.1} m  <- Along-track timing",
        p_rtn[(1, 1)].sqrt()
    );
    println!(
        "  Normal (N):     {:.1} m  <- Cross-track offset",
        p_rtn[(2, 2)].sqrt()
    );

    // Physical interpretation
    println!("\n--- Physical Interpretation ---");
    println!("RTN frame aligns with the orbit:");
    println!("  R (Radial): Points from Earth center to satellite");
    println!("  T (Tangential): Points along velocity direction");
    println!("  N (Normal): Completes right-hand system (cross-track)");
    println!();
    println!("Key insight: Along-track (T) uncertainty grows fastest because");
    println!("velocity uncertainty causes timing errors that accumulate.");
    println!(
        "After one orbit: T/R ratio = {:.1}x",
        p_rtn[(1, 1)].sqrt() / p_rtn[(0, 0)].sqrt()
    );

    // Show correlation structure
    println!("\n--- Position Correlation Matrix (RTN) ---");
    let std_r = p_rtn[(0, 0)].sqrt();
    let std_t = p_rtn[(1, 1)].sqrt();
    let std_n = p_rtn[(2, 2)].sqrt();

    println!("       R      T      N");
    println!(
        "  R  {:6.3} {:6.3} {:6.3}",
        p_rtn[(0, 0)] / (std_r * std_r),
        p_rtn[(0, 1)] / (std_r * std_t),
        p_rtn[(0, 2)] / (std_r * std_n)
    );
    println!(
        "  T  {:6.3} {:6.3} {:6.3}",
        p_rtn[(1, 0)] / (std_t * std_r),
        p_rtn[(1, 1)] / (std_t * std_t),
        p_rtn[(1, 2)] / (std_t * std_n)
    );
    println!(
        "  N  {:6.3} {:6.3} {:6.3}",
        p_rtn[(2, 0)] / (std_n * std_r),
        p_rtn[(2, 1)] / (std_n * std_t),
        p_rtn[(2, 2)] / (std_n * std_n)
    );

    // Validate
    assert_eq!(p_gcrf.nrows(), 6);
    assert_eq!(p_rtn.nrows(), 6);
    assert!(p_rtn[(1, 1)].sqrt() > p_rtn[(0, 0)].sqrt()); // T > R

    println!("\nExample validated successfully!");
}

Output

PythonRust

=== Covariance in RTN Frame ===

Initial position std (ECI): 10.0 m in each axis

--- GCRF Frame Results ---
Position std (X, Y, Z):
  X: 254.3 m
  Y: 27.0 m
  Z: 41.9 m

--- RTN Frame Results ---
Position std (R, T, N):
  Radial (R):     8.7 m  <- Altitude uncertainty
  Tangential (T): 258.8 m  <- Along-track timing
  Normal (N):     10.0 m  <- Cross-track offset

--- Physical Interpretation ---
RTN frame aligns with the orbit:
  R (Radial): Points from Earth center to satellite
  T (Tangential): Points along velocity direction
  N (Normal): Completes right-hand system (cross-track)

Key insight: Along-track (T) uncertainty grows fastest because
velocity uncertainty causes timing errors that accumulate.
After one orbit: T/R ratio = 29.7x

--- Position Correlation Matrix (RTN) ---
       R      T      N
  R   1.000 -0.642  0.000
  T  -0.642  1.000 -0.000
  N   0.000 -0.000  1.000

Example validated successfully!

=== Covariance in RTN Frame ===

Initial position std (ECI): 10.0 m in each axis

--- GCRF Frame Results ---
Position std (X, Y, Z):
  X: 10.0 m
  Y: 184.6 m
  Z: 184.6 m

--- RTN Frame Results ---
Position std (R, T, N):
  Radial (R):     10.0 m  <- Altitude uncertainty
  Tangential (T): 260.8 m  <- Along-track timing
  Normal (N):     10.0 m  <- Cross-track offset

--- Physical Interpretation ---
RTN frame aligns with the orbit:
  R (Radial): Points from Earth center to satellite
  T (Tangential): Points along velocity direction
  N (Normal): Completes right-hand system (cross-track)

Key insight: Along-track (T) uncertainty grows fastest because
velocity uncertainty causes timing errors that accumulate.
After one orbit: T/R ratio = 26.1x

--- Position Correlation Matrix (RTN) ---
       R      T      N
  R   1.000 -0.745 -0.000
  T  -0.745  1.000  0.000
  N  -0.000  0.000  1.000

Example validated successfully!

RTN Frame Interpretation¶

Component	Physical Meaning	Typical Behavior
Radial (R)	Altitude uncertainty	Bounded oscillation
Tangential (T)	Along-track timing	Unbounded growth
Normal (N)	Cross-track offset	Bounded oscillation

The along-track (tangential) uncertainty grows fastest because velocity uncertainty causes timing errors that accumulate over time. After one orbit, the T/R ratio is typically 20-30x.

Covariance Evolution Visualization¶

The following plot shows how position uncertainty evolves over three orbital periods in the ECI frame:

Plot Source

covariance_evolution.py
import numpy as np
import plotly.graph_objects as go

# Add plots directory to path for importing brahe_theme
sys.path.insert(0, str(pathlib.Path(__file__).parent.parent.parent))
from brahe_theme import save_themed_html, get_theme_colors

# Configuration
SCRIPT_NAME = pathlib.Path(__file__).stem
OUTDIR = pathlib.Path(os.getenv("BRAHE_FIGURE_OUTPUT_DIR", "./docs/figures/"))
os.makedirs(OUTDIR, exist_ok=True)

# Initialize EOP data
bh.initialize_eop()

# Create initial epoch and state
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 500e3, 0.01, 45.0, 15.0, 30.0, 45.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Create propagation config with STM enabled and history storage
prop_config = bh.NumericalPropagationConfig.default().with_stm().with_stm_history()

# Create propagator (two-body for clean demonstration)
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.two_body(),
    None,
)

# Define initial covariance (diagonal)
# Position uncertainty: 10 m in each axis (100 m² variance)
# Velocity uncertainty: 0.01 m/s in each axis (0.0001 m²/s² variance)
P0 = np.diag([100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001])

# Propagate for 3 orbital periods
orbital_period = bh.orbital_period(oe[0])
total_time = 3 * orbital_period
prop.propagate_to(epoch + total_time)

# Sample covariance evolution
times = []  # in orbital periods
pos_sigma_r = []  # Radial (x) std dev
pos_sigma_t = []  # Tangential (y) std dev
pos_sigma_n = []  # Normal (z) std dev
pos_total = []  # Total position std dev

dt = orbital_period / 50  # 50 samples per orbit
t = 0.0
while t <= total_time:
    current_epoch = epoch + t
    stm = prop.stm_at(current_epoch)

    if stm is not None:
        # Propagate covariance: P(t) = STM @ P0 @ STM^T
        P = stm @ P0 @ stm.T

        # Extract position standard deviations
        sigma_x = np.sqrt(P[0, 0])
        sigma_y = np.sqrt(P[1, 1])
        sigma_z = np.sqrt(P[2, 2])

        times.append(t / orbital_period)  # Convert to orbital periods
        pos_sigma_r.append(sigma_x)
        pos_sigma_t.append(sigma_y)
        pos_sigma_n.append(sigma_z)
        pos_total.append(np.sqrt(sigma_x**2 + sigma_y**2 + sigma_z**2))

    t += dt


def create_figure(theme):
    colors = get_theme_colors(theme)

    fig = go.Figure()

    # Position uncertainty traces
    fig.add_trace(
        go.Scatter(
            x=times,
            y=pos_sigma_r,
            mode="lines",
            name="X (radial-like)",
            line=dict(color=colors["primary"], width=2),
        )
    )

    fig.add_trace(
        go.Scatter(
            x=times,
            y=pos_sigma_t,
            mode="lines",
            name="Y (along-track-like)",
            line=dict(color=colors["secondary"], width=2),
        )
    )

    fig.add_trace(
        go.Scatter(
            x=times,
            y=pos_sigma_n,
            mode="lines",
            name="Z (cross-track-like)",
            line=dict(color=colors["accent"], width=2),
        )
    )

    fig.add_trace(
        go.Scatter(
            x=times,
            y=pos_total,
            mode="lines",
            name="Total (RSS)",
            line=dict(color=colors["error"], width=2, dash="dash"),
        )
    )

    # Initial uncertainty reference
    initial_total = np.sqrt(3 * 100.0)  # sqrt(3 * 100 m²)
    fig.add_hline(
        y=initial_total,
        line_dash="dot",
        line_color="gray",
        annotation_text=f"Initial: {initial_total:.1f} m",
        annotation_position="top right",
    )

    fig.update_layout(
        title="Position Uncertainty Evolution (Two-Body)",
        xaxis_title="Time (orbital periods)",
        yaxis_title="Position Std Dev (m)",
        showlegend=True,
        legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1),
        height=500,
        margin=dict(l=60, r=40, t=80, b=60),
    )

    return fig


# Save themed HTML files
light_path, dark_path = save_themed_html(create_figure, OUTDIR / SCRIPT_NAME)
print(f"Generated {light_path}")
print(f"Generated {dark_path}")

RTN Covariance Evolution¶

The RTN frame clearly shows why along-track error dominates:

Plot Source

covariance_rtn_evolution.py
import brahe as bh
import numpy as np
import plotly.graph_objects as go

# Add plots directory to path for importing brahe_theme
sys.path.insert(0, str(pathlib.Path(__file__).parent.parent.parent))
from brahe_theme import save_themed_html, get_theme_colors

# Configuration
SCRIPT_NAME = pathlib.Path(__file__).stem
OUTDIR = pathlib.Path(os.getenv("BRAHE_FIGURE_OUTPUT_DIR", "./docs/figures/"))
os.makedirs(OUTDIR, exist_ok=True)

# Initialize EOP data
bh.initialize_eop()

# Create initial epoch and state
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 500e3, 0.01, 45.0, 15.0, 30.0, 45.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Create propagation config with STM enabled and history storage
prop_config = bh.NumericalPropagationConfig.default().with_stm().with_stm_history()

# Define initial covariance (diagonal)
# Position uncertainty: 10 m in each axis (100 m² variance)
# Velocity uncertainty: 0.01 m/s in each axis (0.0001 m²/s² variance)
P0 = np.diag([100.0, 100.0, 100.0, 0.0001, 0.0001, 0.0001])

# Create propagator with initial covariance
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.two_body(),
    None,
    initial_covariance=P0,
)

# Propagate for 3 orbital periods
orbital_period = bh.orbital_period(oe[0])
total_time = 3 * orbital_period
prop.propagate_to(epoch + total_time)

# Sample RTN covariance evolution using STM-based propagation
# This avoids numerical issues with covariance interpolation
times = []  # in orbital periods
sigma_r = []  # Radial std dev
sigma_t = []  # Tangential std dev
sigma_n = []  # Normal std dev

dt = orbital_period / 50  # 50 samples per orbit
t = 0.0
while t <= total_time:
    current_epoch = epoch + t
    stm = prop.stm_at(current_epoch)

    if stm is not None:
        # Propagate covariance in ECI: P(t) = STM @ P0 @ STM^T
        P_eci = stm @ P0 @ stm.T

        # Get state at current epoch to compute RTN rotation
        state_t = prop.state(current_epoch)
        if state_t is not None:
            # Compute RTN rotation matrix from ECI state
            r = state_t[:3]
            v = state_t[3:]

            # RTN basis vectors
            r_hat = r / np.linalg.norm(r)  # Radial
            h = np.cross(r, v)  # Angular momentum
            n_hat = h / np.linalg.norm(h)  # Normal (cross-track)
            t_hat = np.cross(n_hat, r_hat)  # Tangential (along-track)

            # Rotation matrix from ECI to RTN (for position)
            R_eci_to_rtn = np.array([r_hat, t_hat, n_hat])

            # Transform position covariance to RTN
            P_pos_eci = P_eci[:3, :3]
            P_pos_rtn = R_eci_to_rtn @ P_pos_eci @ R_eci_to_rtn.T

            times.append(t / orbital_period)
            sigma_r.append(np.sqrt(P_pos_rtn[0, 0]))
            sigma_t.append(np.sqrt(P_pos_rtn[1, 1]))
            sigma_n.append(np.sqrt(P_pos_rtn[2, 2]))

    t += dt


def create_figure(theme):
    colors = get_theme_colors(theme)

    fig = go.Figure()

    # RTN uncertainty traces
    fig.add_trace(
        go.Scatter(
            x=times,
            y=sigma_r,
            mode="lines",
            name="Radial (R)",
            line=dict(color=colors["primary"], width=2),
        )
    )

    fig.add_trace(
        go.Scatter(
            x=times,
            y=sigma_t,
            mode="lines",
            name="Tangential (T)",
            line=dict(color=colors["secondary"], width=2),
        )
    )

    fig.add_trace(
        go.Scatter(
            x=times,
            y=sigma_n,
            mode="lines",
            name="Normal (N)",
            line=dict(color=colors["accent"], width=2),
        )
    )

    # Add annotations for physical interpretation
    fig.add_annotation(
        x=2.5,
        y=sigma_t[-1] * 0.9,
        text="Along-track: unbounded growth",
        showarrow=False,
        font=dict(size=10),
    )

    fig.add_annotation(
        x=2.5,
        y=sigma_r[-1] * 1.5,
        text="Radial/Normal: bounded oscillation",
        showarrow=False,
        font=dict(size=10),
    )

    # Initial uncertainty reference
    initial_std = 10.0  # m
    fig.add_hline(
        y=initial_std,
        line_dash="dot",
        line_color="gray",
        annotation_text=f"Initial: {initial_std:.0f} m",
        annotation_position="top left",
    )

    fig.update_layout(
        title="Position Uncertainty Evolution in RTN Frame",
        xaxis_title="Time (orbital periods)",
        yaxis_title="Position Std Dev (m)",
        showlegend=True,
        legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1),
        height=500,
        margin=dict(l=60, r=40, t=80, b=60),
    )

    return fig


# Save themed HTML files
light_path, dark_path = save_themed_html(create_figure, OUTDIR / SCRIPT_NAME)
print(f"Generated {light_path}")
print(f"Generated {dark_path}")

Sensitivity Propagation¶

In addition to STM propagation (which tracks sensitivity to initial state), the propagator can compute parameter sensitivity - how the state changes with respect to configuration parameters.

Parameter Sensitivity¶

The sensitivity matrix \(S(t)\) describes how the state at time \(t\) depends on the parameters:

\[S(t) = \frac{\partial \mathbf{x}(t)}{\partial \mathbf{p}}\]

where \(\mathbf{p}\) is the parameter vector. This enables answering questions like:

"How does a 1% uncertainty in drag coefficient affect position prediction?"
"What is the impact of mass uncertainty on orbit determination?"

Configuration Parameters¶

The default parameter vector contains spacecraft physical properties:

Index	Parameter	Units	Description
0	mass	kg	Spacecraft mass
1	drag_area	m²	Cross-sectional area for drag
2	Cd	-	Drag coefficient
3	srp_area	m²	Cross-sectional area for SRP
4	Cr	-	Solar radiation pressure coefficient

Enabling Sensitivity Propagation¶

PythonRust

import numpy as np
import brahe as bh

# Initialize EOP and space weather data (required for NRLMSISE-00 drag model)
bh.initialize_eop()
bh.initialize_sw()

# Create initial epoch and state
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 400e3, 0.01, 45.0, 0.0, 0.0, 0.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Create propagation config with sensitivity enabled
prop_config = (
    bh.NumericalPropagationConfig.default()
    .with_sensitivity()
    .with_sensitivity_history()
)

# Define spacecraft parameters: [mass, drag_area, Cd, srp_area, Cr]
params = np.array([500.0, 2.0, 2.2, 2.0, 1.3])

# Create propagator with full force model (needed for parameter sensitivity)
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.default(),
    params=params,
)

print("Spacecraft Parameters:")
print(f"  Mass: {params[0]:.1f} kg")
print(f"  Drag area: {params[1]:.1f} m²")
print(f"  Drag coefficient (Cd): {params[2]:.1f}")
print(f"  SRP area: {params[3]:.1f} m²")
print(f"  SRP coefficient (Cr): {params[4]:.1f}")

# Propagate for one orbital period
orbital_period = bh.orbital_period(oe[0])
prop.propagate_to(epoch + orbital_period)

# Get the sensitivity matrix (6 x 5)
sens = prop.sensitivity()

if sens is not None:
    print(f"\nSensitivity Matrix shape: {sens.shape}")
    print(
        "(Rows: state components [x,y,z,vx,vy,vz], Cols: params [mass,A_d,Cd,A_s,Cr])"
    )

    # Analyze position sensitivity to each parameter
    pos_sens = sens[:3, :]  # First 3 rows
    param_names = ["mass", "drag_area", "Cd", "srp_area", "Cr"]

    print("\nPosition sensitivity magnitude to each parameter:")
    for i, name in enumerate(param_names):
        # Position sensitivity magnitude for this parameter
        mag = np.linalg.norm(pos_sens[:, i])
        print(f"  {name:10s}: {mag:.3e} m per unit param")

    # Compute impact of 1% parameter uncertainties
    print("\nPosition error from 1% parameter uncertainty:")
    param_uncertainties = params * 0.01  # 1% of each parameter
    for i, name in enumerate(param_names):
        # dpos = sensitivity * dparam
        pos_error = np.linalg.norm(pos_sens[:, i]) * param_uncertainties[i]
        print(f"  {name:10s}: {pos_error:.1f} m")

    # Total position error (RSS)
    total_pos_error = 0.0
    for i in range(len(params)):
        pos_error = np.linalg.norm(pos_sens[:, i]) * param_uncertainties[i]
        total_pos_error += pos_error**2
    total_pos_error = np.sqrt(total_pos_error)
    print(f"\n  Total (RSS): {total_pos_error:.1f} m")

    # Validate
    assert sens.shape == (6, 5)
    print("\nExample validated successfully!")
else:
    print("\nSensitivity not available (may require full force model)")

use brahe as bh;
use bh::traits::DStatePropagator;
use nalgebra as na;
use std::f64::consts::PI;

fn main() {
    // Initialize EOP and space weather data (required for NRLMSISE-00 drag model)
    bh::initialize_eop().unwrap();
    bh::initialize_sw().unwrap();

    // Create initial epoch and state
    let epoch = bh::Epoch::from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh::TimeSystem::UTC);
    let oe = na::SVector::<f64, 6>::new(bh::R_EARTH + 400e3, 0.01, 45.0, 0.0, 0.0, 0.0);
    let state = bh::state_koe_to_eci(oe, bh::AngleFormat::Degrees);

    // Create propagation config with sensitivity enabled
    let mut prop_config = bh::NumericalPropagationConfig::default();
    prop_config.variational.enable_sensitivity = true;
    prop_config.variational.store_sensitivity_history = true;

    // Define spacecraft parameters: [mass, drag_area, Cd, srp_area, Cr]
    let params = na::DVector::from_vec(vec![500.0, 2.0, 2.2, 2.0, 1.3]);

    // Create propagator with full force model (needed for parameter sensitivity)
    let mut prop = bh::DNumericalOrbitPropagator::new(
        epoch,
        na::DVector::from_column_slice(state.as_slice()),
        prop_config,
        bh::ForceModelConfig::default(),
        Some(params.clone()),
        None,
        None,
        None,
    )
    .unwrap();

    println!("Spacecraft Parameters:");
    println!("  Mass: {:.1} kg", params[0]);
    println!("  Drag area: {:.1} m²", params[1]);
    println!("  Drag coefficient (Cd): {:.1}", params[2]);
    println!("  SRP area: {:.1} m²", params[3]);
    println!("  SRP coefficient (Cr): {:.1}", params[4]);

    // Propagate for one orbital period
    let orbital_period = 2.0 * PI * (oe[0].powi(3) / bh::GM_EARTH).sqrt();
    prop.propagate_to(epoch + orbital_period);

    // Get the sensitivity matrix (6 x 5)
    if let Some(sens) = prop.sensitivity() {
        println!(
            "\nSensitivity Matrix shape: ({}, {})",
            sens.nrows(),
            sens.ncols()
        );
        println!("(Rows: state components [x,y,z,vx,vy,vz], Cols: params [mass,A_d,Cd,A_s,Cr])");

        let param_names = ["mass", "drag_area", "Cd", "srp_area", "Cr"];

        println!("\nPosition sensitivity magnitude to each parameter:");
        for (i, name) in param_names.iter().enumerate() {
            // Position sensitivity magnitude for this parameter
            let pos_sens = na::Vector3::new(sens[(0, i)], sens[(1, i)], sens[(2, i)]);
            let mag = pos_sens.norm();
            println!("  {:10}: {:.3e} m per unit param", name, mag);
        }

        // Compute impact of 1% parameter uncertainties
        println!("\nPosition error from 1% parameter uncertainty:");
        let param_uncertainties: Vec<f64> = params.iter().map(|p| p * 0.01).collect();

        let mut total_pos_error_sq = 0.0;
        for (i, name) in param_names.iter().enumerate() {
            let pos_sens = na::Vector3::new(sens[(0, i)], sens[(1, i)], sens[(2, i)]);
            let pos_error = pos_sens.norm() * param_uncertainties[i];
            total_pos_error_sq += pos_error.powi(2);
            println!("  {:10}: {:.1} m", name, pos_error);
        }

        let total_pos_error = total_pos_error_sq.sqrt();
        println!("\n  Total (RSS): {:.1} m", total_pos_error);

        // Validate
        assert_eq!(sens.nrows(), 6);
        assert_eq!(sens.ncols(), 5);
        println!("\nExample validated successfully!");
    } else {
        println!("\nSensitivity not available (may require full force model)");
    }
}

Output

PythonRust

Spacecraft Parameters:
  Mass: 500.0 kg
  Drag area: 2.0 m²
  Drag coefficient (Cd): 2.2
  SRP area: 2.0 m²
  SRP coefficient (Cr): 1.3

Sensitivity Matrix shape: (6, 5)
(Rows: state components [x,y,z,vx,vy,vz], Cols: params [mass,A_d,Cd,A_s,Cr])

Position sensitivity magnitude to each parameter:
  mass      : 9.384e-02 m per unit param
  drag_area : 2.343e+01 m per unit param
  Cd        : 2.130e+01 m per unit param
  srp_area  : 6.119e-02 m per unit param
  Cr        : 9.414e-02 m per unit param

Position error from 1% parameter uncertainty:
  mass      : 0.5 m
  drag_area : 0.5 m
  Cd        : 0.5 m
  srp_area  : 0.0 m
  Cr        : 0.0 m

  Total (RSS): 0.8 m

Example validated successfully!

Spacecraft Parameters:
  Mass: 500.0 kg
  Drag area: 2.0 m²
  Drag coefficient (Cd): 2.2
  SRP area: 2.0 m²
  SRP coefficient (Cr): 1.3

Sensitivity Matrix shape: (6, 5)
(Rows: state components [x,y,z,vx,vy,vz], Cols: params [mass,A_d,Cd,A_s,Cr])

Position sensitivity magnitude to each parameter:
  mass      : 9.384e-2 m per unit param
  drag_area : 2.343e1 m per unit param
  Cd        : 2.130e1 m per unit param
  srp_area  : 6.119e-2 m per unit param
  Cr        : 9.414e-2 m per unit param

Position error from 1% parameter uncertainty:
  mass      : 0.5 m
  drag_area : 0.5 m
  Cd        : 0.5 m
  srp_area  : 0.0 m
  Cr        : 0.0 m

  Total (RSS): 0.8 m

Example validated successfully!

Interpreting Sensitivity Results¶

The sensitivity matrix \(S\) is 6x5 (state dimension x parameter count):

Column 0: Sensitivity to mass - affects drag acceleration (\(a \propto 1/m\))
Column 1: Sensitivity to drag area - affects drag force
Column 2: Sensitivity to Cd - drag coefficient uncertainty
Column 3: Sensitivity to SRP area - solar radiation pressure
Column 4: Sensitivity to Cr - SRP coefficient uncertainty

Physical insights:

For LEO orbits, drag parameters (mass, drag_area, Cd) typically dominate
For GEO orbits, SRP parameters (srp_area, Cr) become more important
Two-body propagation shows zero sensitivity (no force depends on parameters)
Sensitivity grows over time as perturbation effects accumulate

Sensitivity Evolution Visualization¶

The following plot shows how position sensitivity to each parameter evolves over time for a LEO orbit:

Plot Source

sensitivity_evolution.py
import numpy as np
import plotly.graph_objects as go

# Add plots directory to path for importing brahe_theme
sys.path.insert(0, str(pathlib.Path(__file__).parent.parent.parent))
from brahe_theme import save_themed_html, get_theme_colors

# Configuration
SCRIPT_NAME = pathlib.Path(__file__).stem
OUTDIR = pathlib.Path(os.getenv("BRAHE_FIGURE_OUTPUT_DIR", "./docs/figures/"))
os.makedirs(OUTDIR, exist_ok=True)

# Initialize EOP and space weather data
# (Space weather is required for NRLMSISE00 atmospheric model)
bh.initialize_eop()
bh.initialize_sw()

# Create initial epoch and state (LEO orbit with significant drag)
epoch = bh.Epoch.from_datetime(2024, 1, 1, 12, 0, 0.0, 0.0, bh.TimeSystem.UTC)
oe = np.array([bh.R_EARTH + 400e3, 0.01, 45.0, 0.0, 0.0, 0.0])
state = bh.state_koe_to_eci(oe, bh.AngleFormat.DEGREES)

# Create propagation config with sensitivity enabled and history storage
prop_config = (
    bh.NumericalPropagationConfig.default()
    .with_sensitivity()
    .with_sensitivity_history()
)

# Define spacecraft parameters: [mass, drag_area, Cd, srp_area, Cr]
params = np.array([500.0, 2.0, 2.2, 2.0, 1.3])

# Create propagator with full force model (uses NRLMSISE00 for drag)
prop = bh.NumericalOrbitPropagator(
    epoch,
    state,
    prop_config,
    bh.ForceModelConfig.default(),
    params=params,
)

# Propagate for 3 orbital periods
orbital_period = bh.orbital_period(oe[0])
total_time = 3 * orbital_period
prop.propagate_to(epoch + total_time)

# Sample sensitivity evolution
param_names = ["mass", "drag_area", "Cd", "srp_area", "Cr"]
times = []  # in orbital periods
sens_mag = {name: [] for name in param_names}  # Position sensitivity magnitude

dt = orbital_period / 50  # 50 samples per orbit
t = 0.0
while t <= total_time:
    current_epoch = epoch + t
    sens = prop.sensitivity_at(current_epoch)

    if sens is not None:
        times.append(t / orbital_period)  # Convert to orbital periods

        # Compute position sensitivity magnitude for each parameter
        for i, name in enumerate(param_names):
            pos_sens = np.linalg.norm(sens[:3, i])
            sens_mag[name].append(pos_sens)

    t += dt


def create_figure(theme):
    colors = get_theme_colors(theme)

    fig = go.Figure()

    color_map = {
        "mass": colors["primary"],
        "drag_area": colors["secondary"],
        "Cd": colors["accent"],
        "srp_area": colors["error"],
        "Cr": "gray",
    }

    # Add traces for each parameter
    for name in param_names:
        if sens_mag[name]:
            fig.add_trace(
                go.Scatter(
                    x=times,
                    y=sens_mag[name],
                    mode="lines",
                    name=name,
                    line=dict(color=color_map[name], width=2),
                )
            )

    fig.update_layout(
        title="Position Sensitivity to Parameters (LEO, 400 km)",
        xaxis_title="Time (orbital periods)",
        yaxis_title="Position Sensitivity (m per unit param)",
        yaxis_type="log",
        showlegend=True,
        legend=dict(orientation="h", yanchor="bottom", y=1.02, xanchor="right", x=1),
        height=500,
        margin=dict(l=60, r=40, t=80, b=60),
    )

    return fig


# Save themed HTML files
light_path, dark_path = save_themed_html(create_figure, OUTDIR / SCRIPT_NAME)
print(f"Generated {light_path}")
print(f"Generated {dark_path}")

Configuration Reference¶

VariationalConfig Options¶

Option	Type	Default	Description
`enable_stm`	bool	false	Enable State Transition Matrix computation
`enable_sensitivity`	bool	false	Enable parameter sensitivity computation
`store_stm_history`	bool	false	Store STM at each trajectory point
`store_sensitivity_history`	bool	false	Store sensitivity at each trajectory point
`jacobian_method`	DifferenceMethod	Central	Finite difference method for Jacobian
`sensitivity_method`	DifferenceMethod	Central	Finite difference method for sensitivity

DifferenceMethod Options¶

Method	Accuracy	Cost	Description
Forward	O(h)	S+1 evaluations	First-order forward differences
Central	O(h²)	2S evaluations	Second-order central differences (default)
Backward	O(h)	S+1 evaluations	First-order backward differences

Computational Considerations¶

STM and sensitivity computation significantly increase computational cost:

Configuration	State dimension	Memory per step
State only	6	~48 bytes
With STM	42 (6 + 36)	~336 bytes
With Sensitivity	36 (6 + 30)	~288 bytes
With Both	72 (6 + 36 + 30)	~576 bytes

Covariance and Sensitivity¶

Full Example¶

Architecture Overview¶

Configuration Hierarchy¶

Auto-Enable Behavior¶

State Transition Matrices (STM)¶

STM Structure¶

STM Properties¶

Enabling STM Propagation¶

Covariance Propagation¶

Creating a Propagator with Initial Covariance¶

Covariance in RTN Frame¶

RTN Frame Interpretation¶

Covariance Evolution Visualization¶

RTN Covariance Evolution¶

Sensitivity Propagation¶

Parameter Sensitivity¶

Configuration Parameters¶

Enabling Sensitivity Propagation¶

Interpreting Sensitivity Results¶

Sensitivity Evolution Visualization¶

Configuration Reference¶

VariationalConfig Options¶

DifferenceMethod Options¶

Computational Considerations¶

See Also¶