2.3.3. Helical Axes

Helical axes are a way to represent a 3D-movement using only one rotation axis for every time step. The movement is described as a translation along this axis and a rotation around it. This method is sometimes used to analyze the actual joint rotation axis, especially for the tibio-femoral joint [44,45]. This description has also been sometimes used for the patello-femoral joint [46,47]. Consequently, the ideas for the conversion of the rotations are discussed here briefly.

The direction of the rotation axis is given by the helical axis. Thus, the Rodrigues' rotation formula [48] can be used to rotate a vector around any given rotation axis. Matrix representation of a helical axis rotation can be achieved by applying this formula to the unit vectors. The rotated unit vectors give the columns of the associated rotation matrix. For the opposite conversion, the direction of this axis is the eigenvector to the eigenvalue of 1, while the angle can be obtained from the other eigenvalues [49,50]. For more details [51] is recommended.
