Quaternions have been a popular tool in 3D computer graphics for more than 20 years. However, classical quaternions are restricted to the representation of rotations, whereas in graphical applications we typically work with rotation composed with translation (i.e., a rigid transformation). Dual quaternions represent rigid transformations in the same way as classical quaternions represent rotati...

We explain the orthogonal planes split (OPS) of quaternions based on the arbitrary choice of one or two linearly independent pure unit quaternions f ,g. Next we systematically generalize the quaternionic Fourier transform (QFT) applied to quaternion fields to conform with the OPS determined by f ,g, or by only one pure unit quaternion f , comment on their geometric meaning, and establish invers...

We introduce a total order and an absolute value function for dual numbers. The of numbers takes number values, has properties similar to those the real define magnitude quaternion, as number. Based upon these, we extend 1-norm, $$\infty$$ -norm, 2-norm quaternion vectors.

Dual quaternions and dual quaternion interpolation are powerful mathematical tools for the spatial analysis of rigid body motions. In this paper, after a review of some basic results and formulas, it will be presented an attempt to use these tools for the the kinematic modeling of human joints. In particular, the kinematic parameters extracted from experimentally acquired data are compared with...

In this paper, we propose a nonlinear adaptive position and attitude tracking controller for a rigid body that requires no information about the mass and inertia matrix of the body. Moreover, we provide sufficient conditions on the reference trajectory that guarantee mass and inertia matrix identification. The controller is shown to be almost globally asymptotically stable and can handle large ...