Attitude

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....

Euler Angles Euler Parameters (Quaternions) representation A four-element vector with three imaginary and one real component is known as Quaternion. These hypercomplex numbers are optimum for numerical stability and memory load. The Euler parameters are a four-dimensional vector that can be used to represent the orientation of a rigid body. The Euler parameters are defined as: $$ q = \begin{bmatrix} q_0 \\ q_1 \\ q_2 \\ q_3 \end{bmatrix} $$ where $q_0$ is the scalar part and $q_1$, $q_2$, and $q_3$ are the vector part....

Attitude Representation Euler Angles Representation A vector of three angles that represent the attitude of the coordinate frame $ i $ with respect to the coordinate frame $ j $ is called Euler angles. Euler angles are the most commonly used attitude representation because it’s easy to use and understand. One of Euler angles’ obvious advantages is their intuitive representation. $$ \text{Euler angles} = \begin{bmatrix} \phi \\ \theta \\ \psi \end{bmatrix} $$ where $\phi$, $\theta$, and $\psi$ are the rotation angles about the $x$, $y$, and $z$ axes, respectively....

Attitude Attitude representation is a set of coordinates that fully describe a rigid body’s orientation with respect to a reference frame. There are an infinite number of attitude representations, each of which has strengths and weaknesses. Choosing the proper attitude representation depends on the estimation algorithm, type of the moving object (e.g. satellite, spacecraft), type of mission, and reference frame selection. Attitude representation impacts mathematical complexity, geometrical singularities, and operational range, so it’s crucial to choose the proper representation for the objectives....

Introduction Attitude determination and control play a vital role in Aerospace engineering. Most aerial or space vehicles have subsystem(s) that must be pointed to a specific direction, known as pointing modes, e.g., Sun pointing, Earth pointing. For example, communications satellites, keeping satellites antenna pointed to the Earth continuously, is the key to the successful mission. That will be achieved only if we have proper knowledge of the vehicle’s orientation; in other words, the attitude must be determined....