## Orientation parameters

### Rotation matrix

Let $$\mathbf{x}_{a}$$ denote a $$3 \times 1$$ column of coordinates of a vector $$\bar{\mathbf{x}}$$ in some right-handed orthonormal coordinate system $$a.$$ Let $$\mathbf{x}_{b}$$ denote coordinates of the same vector in another right-handed orthonormal coordinate system $$b.$$ Then
$$\mathbf{x}_{a} = \mathbf{R}_{ab} \mathbf{x}_{b},$$
where $$\mathbf{R}_{ab}$$ is a $$3 \times 3$$ matrix. It is usually said that the matrix $$\mathbf{R}_{ab}$$ defines a rotation from the b-frame to the a-frame. Note that
$$\mathbf{R}^{-1}_{ab} = \mathbf{R}_{ba} = \mathbf{R}^{\mathrm{T}}_{ab}, \; \mathbf{R}_{ab}\mathbf{R}^{\mathrm{T}}_{ab} = \mathbf{I}_{3 \times 3}.$$
Here and further on the symbol $$\mathbf{I}$$ denotes an identity matrix of a suitable size. It is possible to show that
$$\dot{\mathbf{R}}_{ab} = \mathbf{R}_{ab}\mathbf{\Omega}_{ab} = \mathbf{R}_{ab}\mathbf{\Omega}_{ib} - \mathbf{\Omega}_{ia}\mathbf{R}_{ab},$$
where $$\mathbf{\Omega}_{ab}$$ is a skew-symmetric matrix of an angular rate of the b-frame relative to the a-frame, and $$\mathbf{\Omega}_{ib}$$ and $$\mathbf{\Omega}_{ia}$$ are skew-symmetric matrices of angular rates of the b- and a-frames relative to some inertial coordinate frame, respectively.

### Euler angles

To be delivered soon

### Quaternions

#### Simple definition of a quaternion

A quaternion can be understood as a pair of a scalar and a vector:
$$\breve{\mathbf{q}} = (q_{0}, \mathbf{q}), \; \mathbf{q} = ( q_{1} \; q_{2} \; q_{3} )^{\mathrm{T}}.$$

#### A product of quaternions

A product of two quaternions $$\breve{\mathbf{q}}$$ and $$\breve{\mathbf{r}}$$ is a quaternion $$\breve{\mathbf{p}}$$, whose scalar and vector parts a determined as
\begin{eqnarray*} \breve{\mathbf{p}} &=& \breve{\mathbf{q}} \ast \breve{\mathbf{r}} = (p_{0}, \mathbf{p}), \\ \nonumber p_{0} &=& q_{0} r_{0} - \mathbf{q}^{\mathrm{T}} \mathbf{r}, \\ \nonumber \mathbf{p} &=& q_{0} \mathbf{r} + r_{0} \mathbf{q} + \mathbf{q} \times \mathbf{r}. \end{eqnarray*}

#### A product of a quaternion and a scalar

$$c \cdot \breve{\mathbf{q}} = (cq_{0}, c\mathbf{q})$$

#### Conjugate quaternion

A conjugate quaternion is defined as
$$\breve{\mathbf{q}}^{\ast} = (q_{0}, -\mathbf{q}).$$

#### Norm of a quaternion

$$\left\Vert \breve{\mathbf{q}} \right\Vert = \sqrt{q_{0}^{2} + q_{1}^{2} + q_{2}^{2} + q_{3}^{2}}$$

#### Inverse quaternion

$$\breve{\mathbf{q}}^{-1} = \frac{\breve{\mathbf{q}}^{\ast}}{\left\Vert \breve{\mathbf{q}} \right\Vert^{2}}$$

#### Representation of rotations by means of quaternions

To be delivered soon