The Moore–Penrose Dual Generalized Inverse Matrix With Application to Kinematic Synthesis of Spatial Linkages

Abstract

The paper initially reports about the properties of an expression of dual generalized inverse matrix currently available in the literature. It is demonstrated that such a matrix does not fulfill all the Penrose conditions. Hence, novel and computationally efficient algorithms/formulas for the computation of the Moore–Penrose dual generalized inverse (MPDGI) are herein proposed. The paper also contains a new algorithm for the singular value decomposition (SVD) of a dual matrix. The availability of these formulas allows the simultaneous solution of overdetermined systems of dual linear equations without requiring the traditional separation in primal and dual parts. This should prove useful for the solution of many kinematic problems. The algorithms/formulas herein deduced have been also tested on the kinematic synthesis of the constant transmission ratio RCCC spatial linkage.

Generalizations of singular value decomposition to dual-numbered matrices

“…Our study of T-SVD is motivated by recent research [5][6][7][8] in mechanics, where the authors consider either a form of SVD that is essentially T-SVD, or a form of Polar Decomposition that is essentially T-Polar Decomposition. For completeness, we describe T-Polar Decomposition below.…”

“…One of the main applications of the T-SVD is in finding the Moore-Penrose generalized inverse of a dual matrix (whenever it exists). This has applications in kinematic synthesis (see [7]).…”

Generalisations of Singular Value Decomposition to dual-numbered matrices

“…Recently in [7], [3], [4] and [8], the authors consider either a form of SVD that is essentially R-SVD, or a form of Polar Decomposition that is essentially R-Polar Decomposition. In each of these cases, the motivation for doing this comes from applications to mechanics.…”

“…In [8], an attempt is made to reduce the problem of finding the R-SVD to solving a system of equations. No proof is given in [8] that this system always has a solution.…”

Reflections Over the Dual Ring—Applications to Kinematic Synthesis

“…Perez-Garcia and McCarthy also applied dual quaternions to the synthesis of spatial linkages, including the RPRP chain [20,21]. More recently, Pennestrì et al proposed a dual version of the Moore-Penrose Generalized Inverse and used it for the synthesis of spatial four-bar linkages [22]. Dual quaternions were also used by Hegedüs et al to factor polynomials in an algorithm for linkage synthesis [23].…”