Quaternion

• What is a Quaternion?
  • Quaternion is a method of expressing attitude, which is often used in the aerospace field.
  • For more information, please refer to related books and the web page.

Movement and Rotation of Astrobee aboard Kibo

Astrobee operates within a coordinate system defined by NASA known as the Local Vertical Local Horizontal (LVLH) frame. This coordinate system is essential for understanding how the robot’s position and orientation are described in relation to its environment, specifically within the International Space Station (ISS).

LVLH Coordinate System

The LVLH coordinate system is a type of body-fixed coordinate system used in orbital mechanics to describe the orientation of a spacecraft relative to its orbit. Here’s how it works:

• X-axis (Forward): Points in the direction of the spacecraft’s velocity vector, which is tangential to the orbit. For the ISS, this axis points roughly in the direction the station is moving along its orbit (ram direction).

• Y-axis (Outboard): Points towards the right side of the spacecraft when looking in the direction of the X-axis. This is perpendicular to the X-axis and generally points outwards from the center of the orbit (starboard direction).

• Z-axis (Nadir): Points towards the center of the Earth (the local vertical). This axis is perpendicular to both the X and Y axes and points downwards from the perspective of the station.

Attitude in LVLH Frame

The attitude of Astrobee in the LVLH frame describes its orientation with respect to these axes:

• Roll (about X-axis): Rotation around the forward direction. Positive roll would tilt the top of the Astrobee to the right (from the robot’s perspective).

• Pitch (about Y-axis): Rotation around the outboard direction. Positive pitch would tilt the front of the Astrobee upwards (towards space).

• Yaw (about Z-axis): Rotation around the nadir direction. Positive yaw would rotate the front of the Astrobee to the right.

Quaternions

Quaternions are a mathematical representation used to describe orientations and rotations in three-dimensional space. Unlike Euler angles (roll, pitch, yaw), quaternions avoid issues such as gimbal lock and provide smoother interpolation of rotations.

A quaternion is composed of four components: $q=(w,x,y,z)$, where:

• $w$ is the scalar part.
• $x,y,z$ are the vector parts.

Quaternions represent a rotation around a unit vector. For example, a quaternion representing a 90-degree rotation around the Z-axis would be $q=(\cos (45^{\circ }),0,0,\sin (45^{\circ }))$.

We can use quaternions to represent rotation with this arbitrary direction as the rotation axis

A unit vector $\lambda =({\lambda }_x,{\lambda }_y,{\lambda }_z)$ represents the rotation axis, and a quaternion $q$ represents the rotation by an angle $\theta$ radians as follows:

$$q=({\lambda }_x\sin (\theta /2),{\lambda }_y\sin (\theta /2),{\lambda }_z\sin (\theta /2),\cos (\theta /2))$$

Here:

• ${\lambda }_x,{\lambda }_y,{\lambda }_z$ are the components of the unit vector $\lambda$ along the x, y, and z axes, respectively.
• $\theta$ is the rotation angle in radians.
• $\sin (\theta /2)$ and $\cos (\theta /2)$ are trigonometric functions that scale the vector and scalar parts of the quaternion.

In practical terms, when programming the Astrobee robot:

• X (Forward/Backward): Moving the robot forward (positive X) or backward (negative X) within the ISS.
• Y (Right/Left): Moving the robot to the right (positive Y) or to the left (negative Y) when facing forward.
• Z (Up/Down): Moving the robot upwards (positive Z) or downwards (negative Z) relative to the Earth’s surface.

**Example: **If you want Astrobee to move to a specific point and face a certain direction, you would specify the coordinates in the LVLH system and the orientation using roll, pitch, and yaw or quaternions.

For instance, to move the Astrobee to a point $(X=1.0,Y=0.5,Z=-0.2)$ and make it face forward with a slight tilt upwards, you might set:

• Position: $X=1.0$, $Y=0.5$, $Z=-0.2$
• Orientation (Euler angles): Roll = $0^{\circ }$, Pitch = $10^{\circ }$, Yaw = $0^{\circ }$
• Orientation (Quaternion): You would calculate the corresponding quaternion for this orientation.

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