Reflections Over the Dual Ring—Applications to Kinematic Synthesis

Least-square problems arise in multiple application areas. The numerical algorithm intended to compute offline the minimum (Euclidian)-norm approximation to an overdetermined system of linear equations, the core of least squares, is based on Householder reflections. It is self-understood, in the application of this algorithm, that the coefficient matrix is dimensionally homogeneous, i.e., all its entries bear the same physical units. Not all applications lead to such matrices, a case in point being parameter identification in mechanical systems involving rigid bodies, whereby both rotation and translation occur; the former being dimensionless and the latter bearing units of length. Because of this feature, dual numbers have found extensive applications in these fields, as they allow the analyst to include translations within the same relations applicable to rotations, on dualization2 of the rotation equations, as occurring in the geometric, kinematic, or dynamic analyses of mechanical systems. After recalling the basic background on dual numbers and introducing reflection matrices defined over the dual ring, we obtain the dual version of Householder reflections applicable to the offline implementation of parameter identification. For the online parameter identification, recursive least squares are to be applied. We provide also the dual version of recursive least squares. Numerical examples are included to illustrate the underlying principles and algorithms.