Dual Orthogonal Matrices in Manipulator

The angular or linear displacements required at each of the joints of a robot manipulator to attain a specified position and orientation of its hand are obtained by solving the closure equations of the manipulator. These equations express the fact that the sequence of coordinate transformations between each link of the manipulator from its hand to its base must equal the transformation from the hand to the base directly. It is well known that the 4 X 4 homogeneous form of the point coordinate transformation matrix together with special coor-dinate frames adapted to the structure of a manipulator yield a standard form of these coordinate transformation matrices known as the Denavit-Hartenberg matrix (Paul 1981). It is not so well known that the transformation equations of the coordinates of lines in these special coordinate frames may b