A Theory of Generalized Inverses Applied to Robotics

Robotics research has made extensive use of techniques based on the Moore-Penrose inverse, or generalized-inverse, of matrices. Recently, it has been pointed out how non-invariant results may, in general, be obtained applying these techniques to some areas of robotics, namely hybrid control and inverse velocity kinematics. Unfortunately, the problems are not restricted to just these particular areas in robotics, but, rather, are connected with misleading definitions of the metric properties of the six dimensional wrench and twist vector spaces used in robotics. The current definitions lead to inconsistent results, i.e. results which are not invariant with respect to changes in the reference frame and/or changes in the dimensional units used to express the problem. As a matter of fact, given a linear system u = A x, where the matrix A may be singular, the Moore-Penrose theory of generalized-inverses may be properly and directly applied only when the vector space U of vectors u and the vector space X of vectors x are inner product spaces. Arbitrary assignment of Euclidean inner products to the space U and X when the vectors u and x have elements with different physical units can lead to inconsistent and non-invariant results. In this paper, the problem of inconsistent, non-invariant solutions xs to u = A x in robotics is briefly reviewed and a general theory for computing consistent, gauge invariant solutions to non-homogeneous systems of the form u = A x is developed. In addition, the dual relationship between rigid-body kinematics and statics is defined formally as a particular, linear algebraic system whose solution system is also a dual system. Examples illustrate the theory.