What is Attitude?

An attitude representation allows the orientation of an object to be mathematically described relative to a reference frame. Two reference frames are defined:

• World frame $\mathcal{F}^w$ — static and unchanging (e.g. inertial or Earth frame)
• Body frame $\mathcal{F}^b$ — attached to and moving with the object

Two conventions exist for the attitude $\Phi$:

$$\Phi_1: \mathcal{F}^w \rightarrow \mathcal{F}^b \qquad \text{(rotation of body frame relative to world frame)}$$$$\Phi_2: \mathcal{F}^b \rightarrow \mathcal{F}^w \qquad \text{(rotation of world frame relative to body frame)}$$

The convention $\Phi_1$ is adopted throughout this series, as it is most aligned with standard GNC engineering practice.

The Three Representations

Three primary methods are used to encode attitude in GNC applications. Each makes a different set of trade-offs:

Property	DCM	Euler Angles	Quaternion
Parameters	9 (3×3 matrix)	3	4
Interpretability	Low	High	Medium
Singularities	None	Yes (gimbal lock)	None
Numerical stability	Requires re-orthogonalisation	Moderate	High
Kinematics integration	Direct	Moderate	Efficient
Interpolation	Moderate	Poor	Smooth (SLERP)
Unique representation	Yes	Yes	No ($\mathbf{q}$ and $-\mathbf{q}$ same rotation)

No single representation is best for all situations. In practice:

• Euler angles are used for display, human interpretation, and initialisation
• DCMs are used when transformation of vectors is the primary operation
• Quaternions are the preferred choice for propagation and integration due to their numerical efficiency and absence of singularities

In This Series