Screw Theory - Coordinate Transformation of Screws

Coordinate Transformation of Screws

The coordinate transformations for screws are easily understood by beginning with the coordinate transformations of the Plücker vector of line, which in turn are obtained from the transformations of the coordinate of points on the line.

Let the displacement of a body be defined by D=(, d), where is the rotation matrix and d is the translation vector. Consider the line in the body defined by the two points p and q, which has the Plücker coordinates,

then in the fixed frame we have the transformed point coordinates P=p+d and Q=q+d, which yield.

Thus, a spatial displacement defines a transformation for Plücker coordinates of lines given by

The matrix is the skew symmetric matrix that performs the cross product operation, that is y=d×y.

The 6×6 matrix constructed from obtained from the spatial displacement D=(, d) can be assembled into the dual matrix

which operates on a screw s=(s.v) to obtain,

The dual matrix =(, ) has determinant 1 and is called a dual orthogonal matrix.