Dynamics: Gravity Gradient

Description

The gravity gradient effector is a component that can be added to a spacecraft that applies a first-order gravity gradient torque acting on the spacecraft. It works with multiple gravitational objects and depending on the centre of mass and mass properties of the spacecraft, applies a non-uniform torque on the craft.

Example Use Cases

• Realistic Gravitation Torques: For larger spacecraft, the gravity gradient effector can pull parts of the spacecraft with a larger force than the remainder of the craft, causing torque.
• Torque Damping: Utilization of the spacecraft geometry to dampen the rotation of a spacecraft over time using a gravity gradient.

Module Implementation

The gravitational gradient torque is only applied on a non-uniform body. The body must have a non-zero centre of mass vector \(\textbf{R}_{BtB}\) and a non-zero total moment of inertia tensor \(I_{Bt}\), about the centre of mass. Here, \(B\) is the body frame and \(Bt\) is the centre of mass frame, where \(\textbf{R}_{BtB}\) represents the vector difference between the origin point of reference on the spacecraft and the centre of mass in the \(B\) frame. It is assumed that a planet centre inertial position vector in the inertial frame \(N\) is given by \(\textbf{R}_{PiN}\), for planet \(i\), then if there are \(n\) planets that contribute to the net gravity, the following equation can determine the torque \(\textbf{L}_B\) that is applied in the body frame:

\[ \textbf{L}_B = \sum_{i=1}^n{3 \mu_i \over |\textbf{R}_{BtPi}|^3 } \cdot I_{Bt} \textbf{R}_{BtPi} \]

Here, \(\mu_i\) is the gravitational parameter constant for the body \(i\) and \(\textbf{R}_{BtPi}\) is the vector from the centre of mass of the spacecraft to the planet’s inertial vector. In this case,

\[ \textbf{R}_{BtPi} = \textbf{R}_{BtN} - \textbf{R}_{PiN} \]

which are the centre of mass and planet inertial vectors respectively. The inertia tensor \(I_{Bt}\) can be calculated by:

\[ I_{Bt} = I_{B} - m_t\tilde{C}\tilde{C}^T \]

where \(I_B\) is the moment of inertia about the origin, \(m_t\) is the total mass of the spacecraft and \(\tilde{C}\) marks the skew matrix calculated from the centre of mass vector \(\textbf{R}_{BtB}\) in the body frame. The body frame torque \(\textbf{L}_B\) is applied to the centre of mass \(\textbf{R}_{BtB}\) of the spacecraft on the next update frame.

Assumptions/Limitations

• The gravity gradient will always act on the total centre of mass, mass and moment of inertia of the spacecraft.