Gravity Gradient Effector

SWITCH TO API

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 center of mass and mass properties of the spacecraft, applies a non-uniform torque on the craft.

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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.

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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 ${\text{R}}_{BtB}$ 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 ${\text{R}}_{BtB}$ 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 ${\text{R}}_{PiN}$, for planet $i$, then if there are $n$ planets that contribute to the net gravity, the following equation can determine the torque ${\text{L}}_B$ that is applied in the body frame:

$${\text{L}}_B=\sum _{i=1}^n\frac{3{\mu }_i}{|{\text{R}}_{BtPi}{|}^3}\cdot I_{Bt}{\text{R}}_{BtPi}$$

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

$${\text{R}}_{BtPi}={\text{R}}_{BtN}-{\text{R}}_{PiN}$$

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 ${\text{R}}_{BtB}$ in the body frame. The body frame torque ${\text{L}}_B$ is applied to the centre of mass ${\text{R}}_{BtB}$ of the spacecraft on the next update frame.

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Assumptions/Limitations

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