# Attitude Representation - Other

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:

$$g_1 = \frac{R_{23}-R_{32}}{1+R_{11}+R_{22}+R_{33}}$$
$$g_2 = \frac{R_{31}-R_{13}}{1+R_{11}+R_{22}+R_{33}}$$
$$g_3 = \frac{R_{12}-R_{21}}{1+R_{11}+R_{22}+R_{33}}$$
where $R_{ij}$ is the element of the rotation matrix.

The Rodrigues Parameter preferred as an attitude error representation because it is a unit vector and it is easy to calculate the attitude error between two quaternions. The attitude error between two quaternions can be calculated using the following:

$$\mathbf{g}_{error} = \frac{2\mathbf{q}_{v}}{q_0}$$
The Gibbs vector components expereince a singularity at $\theta = \pi$, which is the same as the Euler angles. The Gibbs vector is not a good representation for small rotations.

## Modified Rodrigues Parameters

In attitude filter design Modified Rodrigues Parameters (MRP) is preferred for attitude error representation. The MRP is a set of three parameters denoted by $\mathbf{m}$ and can be directly derived from axis-angle $(e, \theta)$ or quaternion representation as follows:

$$\mathbf{m} = \frac{\mathbf{q}_{v}}{1+q_0}$$
$$\mathbf{m} = \frac{e \sin\frac{\theta}{2}}{1+\cos\frac{\theta}{2}}$$
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.

Due to above equation the maximum equivalent rotation to describe is $\pm 360^{\circ}$ (the singularity occurs in $\pm 360^{\circ}$).

## Cayley-Klein

The Cayley-Klein parameters are consisting of 4 parameters which are closely related to the quaternions and denoted by matrix $\mathbf{K}_{2\times 2}$.

$$K = \begin{bmatrix} \alpha & \beta \\ \gamma & \sigma \end{bmatrix}$$
and satisfy the constraints
$$\alpha \bar{\alpha} + \gamma \bar{\gamma} = 1 \\ \alpha \bar{\alpha} + \beta \bar{\beta} = 1 \\ \alpha \bar{\beta} + \gamma \bar{\sigma} = 0 \\ \alpha \sigma + \beta \gamma = 1 \\ \beta = -\bar{\gamma} \\ \sigma = \bar{\alpha}$$
where $\alpha$, $\beta$, $\gamma$, and $\sigma$ are the Cayley-Klein parameters and $\bar{\alpha}$, $\bar{\beta}$, $\bar{\gamma}$, and $\bar{\sigma}$ are the conjugate of the Cayley-Klein parameters.

The corresponding quaternions are defined_as:

$$\mathbf{q}_K = \begin{bmatrix} \frac{ \alpha + \sigma }{2} \\ \frac{ -i(\beta + \gamma) }{2} \\ \frac{ \beta - \gamma }{2} \\ \frac{ -i(\alpha - \sigma) }{2} \end{bmatrix}$$

## References:

[1] Markley, F. Landis, and John L. Crassidis. Fundamentals of spacecraft attitude determination and control. Vol. 1286. New York, NY, USA:: Springer New York, 2014.
[2] Junkins, John L., and Hanspeter Schaub. Analytical mechanics of space systems. American Institute of Aeronautics and Astronautics, 2009.
[3] De Ruiter, Anton H., Christopher Damaren, and James R. Forbes. Spacecraft dynamics and control: an introduction. John Wiley & Sons, 2012.
[4] Wertz, James R., ed. Spacecraft attitude determination and control. Vol. 73. Springer Science & Business Media, 2012.
[5] Vepa, Ranjan. Dynamics and Control of Autonomous Space Vehicles and Robotics. Cambridge University Press, 2019.
[6] Shuster, Malcolm D. “A survey of attitude representations.” Navigation 8.9 (1993): 439-517.
[7] Markley, F. Landis. “Attitude error representations for Kalman filtering.” Journal of guidance, control, and dynamics 26.2 (2003): 311-317.
[8] Markley, F. Landis, and Frank H. Bauer. Attitude representations for Kalman filtering. No. AAS-01-309. 2001.

##### Arman Asgharpoor Golroudbari
###### Space-AI Researcher

My research interests revolve around planetary rovers and spacecraft vision-based navigation.