Specification for Relative Orbit propagation

1. Overview

1. functions

  • The RelativeOrbit class calculates the position of a satellite with respect to a reference satellite. This class calculates the position both in the LVLH frame and inertial frame. Users can choose the update method between:
    • relative dynamics propagation using RK4
    • update using STM(State Transition Matrix)
  • 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/relative_orbit.hpp, .cpp
    • Definition of the class
  • Libraries
    • src/library/orbit/relative_orbit_models.hpp, .cpp
    • Library to store equations for various relative dynamics

3. How to use

  • Relative orbit propagation is available only when multiple satellites are simulated.

    • The sample case in S2E_CORE simulates a single satellite. For an example of simulating multiple satellites, please refer to the tutorial.

  • Confirm the instance of RelativeInformation is the member of each satellite.

  • Set up the configuration of the [ORBIT] section in the sample_spacecraft.ini.

    • Set propagate_mode = RELATIVE to use the relative orbit propagation
    • Choose relative_orbit_update_method.
      • relative_orbit_update_method = 0 means the orbit is updated with the propagation of the relative dynamics equation( $\dot{\boldsymbol{x}}=\boldsymbol{Ax}+\boldsymbol{Bu}$ , i.e., Hill equation).
      • relative_orbit_update_method = 1 means the orbit is updated with the STM( $\boldsymbol{x}(t)=\boldsymbol{\Phi}(t,t_0)\boldsymbol{x}(t_0)$ , i.e., Clohessy-Wiltshire solution).
    • When you choose relative_orbit_update_method = 0, set relative_dynamics_model_type.
    • When you choose relative_orbit_update_method = 1, set stm_model_type.
    • Set the initial relative position of the satellite in the LVLH frame. LVLH frame definition is:
      • $\boldsymbol{x}$ is a direction vector from the reference satellite ("chief" in the figure) radially outward.
      • The direction of $\boldsymbol{z}$ corresponds to the angular momentum vector of the reference satellite orbit.
      • The direction of $\boldsymbol{y}$ is determined by $\boldsymbol{z}\times\boldsymbol{x}$.
    Definition of LVLH frame [1]
    • Set the ID and ini file name of the reference satellite.
      • NOTE: Confirm the propagate_mode of the reference satellite is not 2. The orbit of the reference satellite must be propagated by the methods other than the relative orbit propagation.

4. How to add a new relative dynamics model

  1. New Relative Dynamics equation
  • Add the name of the dynamics model to the RelativeOrbitModel enum in relative_orbit_models.hpp.
  • Add the function to calculate the system matrix like CalcHillSystemMatrix in relative_orbit_models.hpp.
  • Edit the CalculateSystemMatrix function in relative_orbit.hpp.
  1. New STM
  • Add the name of the dynamics model to StmModel enum in relative_orbit_models.hpp.
  • Add the function to calculate the system matrix as CalcHcwStmin relative_orbit_models.hpp.
  • Edit the CalculateStm function in relative_orbit.hpp.

2. Explanation of Algorithm

1. InitializeState

1. overview

  • The InitializeState function initializes the orbit of the satellite.

2. inputs and outputs

  • input
    • relative_position_lvlh_m, relative_velocity_lvlh_m_s
    • The initial state of the satellite
    • gravity_constant_m3_s2
      • The gravity constant of the reference celestial body $\mu$
    • initial_time_s
      • Initial simulation time (default value is 0)
  • output
    • none

3. algorithm

4. note

2. CalculateSystemMatrix

1. overview

  • The CalculateSystemMatrix function is used only inside the RelativeOrbit class. This function calls the system matrix calculation function according to relative_dynamics_model_type.

2. inputs and outputs

  • input
    • relative_dynamics_model_type
      • The type of relative dynamics model
    • reference_sat_orbit
      • The orbit of the reference satellite
    • gravity_constant_m3_s2
      • The gravity constant $\mu$
  • output
    • none

3. algorithm

4. note

3. CalculateStm

1. overview

  • The CalculateStm function is used only inside the RelativeOrbit class. This function calls the system matrix calculation function according to stm_model_type.

2. inputs and outputs

  • input
    • stm_model_type
      • The type of relative dynamics model
    • reference_sat_orbit
      • The orbit of the reference satellite
    • gravity_constant_m3_s2
      • The gravity constant $\mu$
    • elapsed_sec
      • Elapsed simulation time
  • output
    • none

3. algorithm

4. note

4. CalculateHillSystemMatrix

1. overview

  • The CalculateHillSystemMatrix function calculates the system matrix of the Hill equation.
  • This function is declared in relative_orbit_models.hpp and is used by the

2. inputs and outputs

  • input
    • orbit_radius
      • Radius of the reference satellite orbit $R$
    • gravity_constant_m3_s2
      • The gravity constant $\mu$
  • output
    • system_matrix
      • system matrix

3. algorithm

  • The mean motion of the reference orbit ($n$) is calculated as follows:

\[ n=\sqrt{\frac{\mu}{R^3}} \]

  • Then, the system matrix ($\boldsymbol{A}$) is calculated as follows:

\[ \boldsymbol{A}= \begin{pmatrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 3n^2 & 0 & 0 & 0 & 2n & 0 \\ 0 & 0 & 0 & -2n & 0 & 0 \\ 0 & 0 & -n^2 & 0 & 0 & 0 \\ \end{pmatrix} \]

4. note

5. CalculateHcwStm

1. overview

  • The CalculateHcwStm function calculates the Hill-Clohessy-Wiltshire STM.
  • This function is declared in relative_orbit_models.hpp and is used by the

2. inputs and outputs

  • input
    • orbit_radius
      • Radius of the reference satellite orbit $R$
    • gravity_constant_m3_s2
      • The gravity constant $\mu$
    • elapsed_sec
      • Elapsed simulation time
  • output
    • system_matrix
      • system matrix

3. algorithm

  • The mean motion of the reference orbit ($n$) is calculated as follows:

\[ n=\sqrt{\frac{\mu}{R^3}} \]

  • Then, the system matrix ($\boldsymbol{A}$) is calculated as follows:

\[ \boldsymbol{\Phi}(t,t0)= \begin{pmatrix} 4-3\cos(nt) & 0 & 0 & \frac{\sin(nt)}{n} & \frac{2}{n}-\frac{2\cos(nt)}{n} & 0 \\ -6nt+6\sin(nt) & 1 & 0 & -\frac{2}{n}+\frac{2\cos(nt)}{n} & \frac{4\sin(nt)}{n}-3t & 0 \\ 0 & 0 & \cos(nt) & 0 & 0 & \frac{\sin(nt)}{n} \\ 3n\sin(nt) & 0 & 0 & cos(nt) & 2\sin(nt) & 0 \\ -6n+6n\cos(nt) & 0 & 0 & -2\sin(nt) & -3+4\cos(nt) & 0 \\ 0 & 0 & -n\sin(nt) & 0 & 0 & \cos(nt) \\ \end{pmatrix} \]

4. note

3. Results of verifications

1. Relative Orbit Propagation

1. overview
  • Make sure the relative orbit is correctly propagated using a simple orbit
2. conditions for the verification
  • input files
    • Initialization files for the two satellites were prepared.
      • satellite0.ini
      • satellite1.ini
  • initial values

    • The orbit of the reference satellite (satellite0) is GEO. The initial position of the satellite (satellite1) is $(0\mathrm{m}, 100\mathrm{m}, 0\mathrm{m})^\mathrm{T}$ in LVLH frame. The orbit was propagated for 86400 sec (the period of GEO).

    • satellite0.ini

      propagate_mode = RK4

//Information used for orbital propagation by the Runge-Kutta method///////////
//initial satellite position[m] 
//＊The coordinate system is defined in PlanetSelect.ini
initial_position_i_m(0) = 4.2164140100E+07  //radius of GEO
initial_position_i_m(1) = 0
initial_position_i_m(2) = 0
//initial satellite velocity[m/s]
//＊The coordinate system is defined in PlanetSelect.ini
initial_velocity_i_m_s(0) = 0
initial_velocity_i_m_s(1) = 3.074661E+03  //Speed of a spacecraft in GEO
initial_velocity_i_m_s(2) = 0
///////////////////////////////////////////////////////////////////////////////

    • satellite1.ini

      propagate_mode = RELATIVE

//Information used for relative orbit propagation//////////////////////////////
//Relative Orbit Update Method (0 means RK4, 1 means STM)
relative_orbit_update_method = 0
// RK4 Relative Dynamics model type (only valid for RK4 update)
// 0: Hill
relative_dynamics_model_type = 0
// STM Relative Dynamics model type (only valid for STM update)
// 0: HCW
stm_model_type = 0
// Initial satellite position relative to the reference satellite in LVLH frame[m]
// * The coordinate system is defined at [PLANET_SELECTION] in SampleSimBase.ini
initial_relative_position_lvlh_m(0) = 0.0
initial_relative_position_lvlh_m(1) = 100.0
initial_relative_position_lvlh_m(2) = 0.0
// initial satellite velocity relative to the reference satellite in LVLH frame[m/s]
initial_relative_velocity_lvlh_m_s(0) = 0.0
initial_relative_velocity_lvlh_m_s(1) = 0.0
initial_relative_velocity_lvlh_m_s(2) = 0.0
// information of reference satellite
reference_satellite_id = 0
///////////////////////////////////////////////////////////////////////////////
3. results
  • The position of satellite1 seen from satellite0 in the inertia frame was calculated.
  • They correctly move periodically.
  • In this example of verification, the RK4 method is used to propagate relative orbits, but the results were the same when using STM.

4. References

  1. Kapila, V., Sparks, A. G., Buffington, J. M., & Yan, Q. (2000). Spacecraft formation flying: Dynamics and control. Journal of Guidance, Control, and Dynamics, 23(3), 561-564.