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Specification for Orbit Propagation with Encke's Method

1. Overview

1. functions

  • The EnckeOrbitPropagation class calculates the satellite position and velocity with Encke's method, including disturbances and controlled accelerations by the satellite.
  • This orbit propagation mode provides an accurate and efficient orbit calculation with disturbance forces.
  • We can also use it for accurate relative orbit propagation, and the feature will be implemented soon.

2. files

  • src/dynamics/orbit/orbit.hpp, cpp
    • Definition of Orbit base class
  • src/dynamics/orbit/initialize_orbit.hpp, .cpp
    • Make an instance of orbit class.
  • src/dynamics/orbit/encke_orbit_propagation.cpp, .hpp
  • We use KeplerOrbit libraries to calculate the reference orbit.

3. How to use

  • Select propagate_mode = ENCKE in the spacecraft's ini file.
  • Select initialize_mode as you want.
    • DEFAULT : Use default initialize method (RK4 and ENCKE use position and velocity, KEPLER uses init_mode_kepler)
    • POSITION_VELOCITY_I : Initialize with position and velocity in the inertial frame
    • ORBITAL_ELEMENTS : Initialize with orbital elements
  • Set the value of error_tolerance, which decides the threshold for the rectification.

2. Explanation of Algorithm

1. EnckeOrbitPropagation::Initialize function

1. Overview

  • This function generates the initial value of the reference orbit and the difference orbit.

2. Inputs and outputs

  • Input
    • $\mu$ : The standard gravitational parameter of the central body
    • $t$ : Time in Julian day
    • $\boldsymbol{r}_{i}$ : Initial position in the inertial frame
    • $\boldsymbol{v}_{i}$ : Initial velocity in the inertial frame
  • Output
    • The reference orbit
    • The difference is set as zero

3. Algorithm

2. EnckeOrbitPropagation::Propagate function

1. Overview

  • This function is the main algorithm of Encke's method and calculates the orbit of the spacecraft.
  • The method separates the orbit to the reference and the difference. The reference is calculated with the Kepler orbit method as a two-body problem, and the difference is calculated, including the disturbances.
    • $\boldsymbol{r}_{ref}$ : Reference orbit
    • $\boldsymbol{\delta}$ : Difference
  • Please refer to the references to learn the original idea of Encke's method.

2. Inputs and outputs

  • Input
    • $\boldsymbol{a}_d$ : Acceleration
    • $t$ : Current time
  • Output
    • $\boldsymbol{r}_{i}$ : Initial position in the inertial frame
    • $\dot{\boldsymbol{r}}_{i}$ : Initial velocity in the inertial frame

3. Algorithm

  1. Rectification
  • If the norm of the difference is larger than the tolerance, we need to update the reference orbit as the latest orbit information.
  1. Update reference orbit
  • The reference orbit is calculated with the Kepler orbit calculation method.
  1. Propagate the difference
  • Propagate the following differential equation. At this moment, we use the fourth-order Runge-Kutta method as a propagator.
$$\begin{align} \ddot{\boldsymbol{\delta}} &= -\frac{\mu}{r_{ref}^3}(\boldsymbol{\delta}+f(q)\boldsymbol{r})+\boldsymbol{a}_d\\\ f(q) &= q \frac{q^2 + 3q + 3}{(1+q)^{1.5} + 1}\\\ q &= \frac{\boldsymbol{\delta}\cdot(\boldsymbol{\delta}-2\boldsymbol{r}_i)}{r_i} \end{align}$$

3. Results of verifications

1. Comparison with RK4

1. Overview

  • We compared the calculated orbit result between RK4 mode and Kepler mode.
  • In the Kepler mode, we verified the correctness of both initialize mode (ORBITAL_ELEMENTS and POSITION_VELOCITY_I).

2. Conditions for the verification

  • sample_simulation_base.ini
    • The following values are modified from the default. 
      EndTimeSec = 10000
LogOutPutIntervalSec = 5
  • SampleDisturbance.ini
    • The disturbance setting is depending on the simulation case.
      • All disabled or enabled. Other settings are default.
  • SampleSat.ini
    • The following values are modified from the default.
      • propagate_mode is changed for each mode.
      • Orbital elements for Kepler 
        semi_major_axis_m = 6794500.0
eccentricity = 0.0015
inclination_rad = 0.9012
raan_rad = 0.1411
arg_perigee_rad = 1.7952
epoch_jday = 2.458940966402607e6
      • Initial position and velocity (compatible value with the orbital elements) 
        initial_position_i_m(0) = 1791860.131
initial_position_i_m(1) = 4240666.743
initial_position_i_m(2) = 4985526.129
initial_velocity_i_m_s(0) = -7349.913889
initial_velocity_i_m_s(1) = 631.6563971
initial_velocity_i_m_s(2) = 2095.780148
  1. Results

  • The following figure shows the difference between orbit derived with Kepler mode initialized with OE and Encke mode without any disturbances.

    • The error is small (less than 10m), and we confirmed that the Encke propagation mode is correct when the disturbances are zero.

  • The following figure shows the difference between orbit derived with RK4 mode and Encke mode with all disturbances.

    • The error is larger than the non-disturbance case, but the $10^4 [m]$ error between the Encke method and the Cowell method is compatible with the ref[2] when using the RK4 in LEO. We confirmed that the Encke propagation mode is correct when all disturbances are included.

4. References

  • [1] David A. Vallado, "Fundamental of Astrodynamics and Applications, Third Edition", ch.8, 2007.
  • [2] Simon P. Shuster, "A Survey and Performance Analysis of Orbit Propagators for LEO, GEO, and Highly Elliptical Orbits", 2017.