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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

Advanced topics

  • Rotation vector
  • Conversion of various rotational parameters
To learn more about the advanced topics, please contact us!