## 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)