Spec_GravityGradient.md

Specification for Gravity Gradient Torque

1. Overview

1. functions

• This class calculates the gravity gradient torque in the Disturbances class.

2. related files

• gravity_gradient.cpp, gravity_gradient.hpp : Definitions and declarations of the class
• disturbance.ini : Initialization file

3. how to use

• Make an instance of the GravityGradient class in InitializeInstances function in disturbances.cpp
  • Create an instance by using the initialization function InitGravityGradient
• Set the parameters in the disturbance.ini
  • Select ENABLE for calculation and logging

2. Explanation of Algorithm

1. CalcTorque_b_Nm function

1. overview

This function performs disturbance calculation and torque output simultaneously.

2. algorithm

Gravity gradient torque is calculated by the following equations (1).

$$\boldsymbol{T}_{GG} = \cfrac{3 \mu}{R_0^5} \boldsymbol{R}_0 \times (\boldsymbol{I}\cdot \boldsymbol{R}_o)$$

where $\mu$ is the gravitational constant of the Earth in normal case, $R_0$ is the distance between the Earth center and the satellite, $\boldsymbol{R_0}$ is the vector from Earth center to the satellite, $\boldsymbol{I}$ is the inertia tensor of the satellite.
Users can change the $\mu$ for other planet with the constructor.

3. Results of verifications

1. verification of gravity gradient torque

1. overview

• Check that the gravity gradient torque equation is performed correctly

2. conditions for the verification

• PropStepSec: 0.001
• StepTimeSec: 0.1
• EndTimeSec: 300
• Inertia tensor: diag [0.17, 0.1, 0.25]
• Initial Quaternion_i2b: [0,0,0,1]
• Initial torque: [0,0,0]
• Initial angular velocity: [0,0,0]
• Disturbance torque: All Disable except gravity gradient torque

3. results

• The order of the gg torque is $10^7$, which seems a proper value.

4. References

1. 姿勢制御研究委員会, 人工衛星の力学と姿勢ハンドブック, 培風館, 2007. (written in Japanese)