Introduction to attitude coordinates

29 Sep 2019

Attitude coordinates are the set of coordinates that describe the orientation of a rigid body. Technically, these coordinates describe the orientation of a frame of reference attached to the rigid body relative to a reference frame. For example, the attitude of a spacecraft relative to a ground station is actually the orientation of the frame attached to the spacecraft relative to the frame attached to the ground station. Similalry, the attitude of a spacecraft relative to a distant star is the orientation of a frame attached to the spacecraft relative to a frame attached to the star. In this case, the star frame is assumed to be inertial and this arrangement is used to determine spacecraft attitude using a star sensor.

A single spacecraft will have multiple frames defined in its flight dynamics software. For example, there will be a spacecraft frame to describe the attitude of the spacecraft relative to an inertial frame. However, the camera of the spacecraft might have a frame attached to it. This frame will describe the orientation of the camera frame relative to the spacecraft frame.

A set of three parameters can completely define the attitude of an object. However, in practice we use sets that have more than three parameters. Some of the attitude coordinate sets are:

• Direction Cosine Matrix
• Euler angles
• Principle rotation parameters
• Euler parameters (quaternions)
• Classical Rodrigues parameters
• Modified Rodrigues parameters
• Stereographic orientation parameters

Difference between position and attitude coordinates

There is a fundamental difference between position coordinates and attitude coordinates. While the error in position between two frames can grow infinitely large, the error in attitude can never be more than \(180^\circ\).

Four truths about attitude coordinates

1. A minimum of three coordinates is required to describe the relative angular displacement between two reference frames.

2. Any minimal set of three coordinates will contain at least one geometrical orientation where the coordinates are singular, namely at least two coordinates are undefined or not unique.

3. At or near such a geometric singularity, the corresponding kinematic differential equations are also singular.

4. The geometric singularities and associated numerical difficulties can be avoided altogether through regularisation. Redundant sets of four or more coordinates exist that are universally valid.

Redundant sets avoid singularities at the cost of additional coordinates. However, just having more than three coordinates doesn’t guarantee that the set will be non-singular.

References

1. H Schaub and J Junkins. Analytical Mechanics of Space Systems. Published by American Institute of Aeronautics and Astronautics, Inc