Quaternion

## Gibbs Vector / Rodrigues Parameter Representation#

The Gibbs vector also known as Rodrigues Parameter is a set of three parameters denoted by $g$ (or $P$) and can be directly derived from axis-angle $(e, \theta)$ or quaternion representation as follows:

$$\mathbf{g} = \frac{\mathbf{q}_{v}}{q_0}$$
$$\mathbf{g} = \frac{e \sin\frac{\theta}{2}}{\cos\frac{\theta}{2}}$$
where $\mathbf{q}_{v}$ is the vector part of the quaternion, $e$ is the unit vector of the axis of rotation, and $\theta$ is the angle of rotation.

The Gibbs vector is a unit vector that represents the axis of rotation and the magnitude of the vector represents the angle of rotation. The Gibbs vector can be used to represent the rotation matrix $C_{\psi\theta\phi}$ as:

$$C_{\psi\theta\phi} = \begin{bmatrix} 1 - 2(g_2^2 + g_3^2) & 2(g_1g_2 - g_3) & 2(g_1g_3 + g_2) \\ 2(g_1g_2 + g_3) & 1 - 2(g_1^2 + g_3^2) & 2(g_2g_3 - g_1) \\ 2(g_1g_3 - g_2) & 2(g_2g_3 + g_1) & 1 - 2(g_1^2 + g_2^2) \end{bmatrix}$$
where $g_1$, $g_2$, and $g_3$ are the components of the Gibbs vector.

The Gibbs vector components expressed in DCM can be calculated using the following: