Introduction
An attitude representation allows the orientation of an object to be mathematically described relative to a reference frame. This mathematical description is a set of parameters or a transformation that describes the orientation of a reference frame with respect to another reference frame. When we describe the attitude, we two reference frames:
- World/Fixed Reference Frame (denoted by \(\mathcal{F}^w\)): which acts as a static reference that says fixed in location and attitude)
- Body/Rotated Reference Frame (denoted by \(\mathcal{F}^b\)): which is rigidly attached to the object whose attitude we would like to describe relative to the world frame.
Now let \(\mathbf{\Phi}\) represent the attitude of an object or frame such that it maps quantities in one frame to the other frame:
\[ \mathbf{\Phi} : \mathcal{F}^w \leftrightarrow \mathcal{F}^b \]
There are two different conventions that could be adopted, we could define that:
- The attitude \(\mathbf{\Phi}\) expresses the rotation of the rotated frame \(\mathcal{F}^b\) relative to the reference frame \(\mathcal{F}^w\) , such that: \( \mathbf{\Phi}_1: \mathcal{F}^w \rightarrow \mathcal{F}^b \)
- The attitude \(\mathbf{\Phi}\) expresses the rotation of the reference frame \(\mathcal{F}^w\) relative to the rotated frame \(\mathcal{F}^b\) , such that: \( \mathbf{\Phi}_2: \mathcal{F}^b \rightarrow \mathcal{F}^w \)
In either convention, that the attitude can be inverted to find the opposite frame mapping or inverse rotation:
\( \mathbf{\Phi}_1: \mathcal{F}^w \rightarrow \mathcal{F}^b \)
\( \mathbf{\Phi}_1^{-1}: \mathcal{F}^w \leftarrow \mathcal{F}^b \)
For the rest of the page it is assumed that the first convention \(\mathbf{\Phi}: \mathcal{F}^w \rightarrow \mathcal{F}^b\) is used.
Mathematical Representations
There are multiple ways to express and store an attitude representation, with the main three being:
- Direct Cosine Matrix (DCM) – Uses a 3×3 matrix to represent the linear transform mapping from one coordinate frame to another rotated coordinate frame.
- Euler Angles – Uses 3 component rotation angles to parameterize the attitude.
- Quaternion – Uses a unit quaternion to parameterize the attitude.
Advantages and Disadvantages
Each representation method has advantages and disadvantages:
- Ease of User Interpretation and Interaction (can you mentally picture it and describe it)
- Storage (size in memory)
- Numerical Issues (stability, computation performance, uniqueness)
- Integration and Kinematics (how to describe a rotating object or changing attitude)
- Interpolation (computer graphics and animation, smoothly changing between two orientations)