Dual Quaternion - Dual Quaternions and 4x4 Homogeneous Transforms

Dual Quaternions and 4x4 Homogeneous Transforms

It might be helpful, especially in rigid body motion, to represent dual quaternions as homogeneous matrices. As given above a dual quaternion can be written as: where r and d are both quaternions. The r quaternion is known as the real or rotational part and the quaternion is known as the dual or displacement part. A 3 dimensional position vector,

can be transformed by constructing the dual-quaternion representation,

then a transformation by is given by

.

The rotation part can be given by

where is the angle of rotation about axis . The rotation part can be expressed as a 3×3 orthogonal matrix by

R =\begin{pmatrix} r_w^2+r_x^2-r_y^2-r_z^2 &2r_xr_y-2r_wr_z &2r_xr_z+2r_wr_y \\ 2r_xr_y+2r_wr_z &r_w^2-r_x^2+r_y^2-r_z^2 &2r_yr_z-2r_wr_x \\ 2r_xr_z-2r_wr_y &2r_yr_z+2r_wr_x &r_w^2-r_x^2-r_y^2+r_z^2\\ \end{pmatrix}.

The displacement can be written as

.

Translation and rotation combined in one transformation matrix is:

= \begin{pmatrix} & & & \Delta x \\ & R & & \Delta y \\ & & & \Delta z \\ 0 & 0 & 0 & 1 \\ \end{pmatrix}

Where the left upper 3×3 matrix is the rotation matrix we just calculated.

Read more about this topic:  Dual Quaternion

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