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 noninvariant results may, in general, be obtained by applying these techniques to other areas of robotics, namely hybrid control and inverse velocity kinematics. Unfortunately, the problems are not restricted to just these particular areas in robotics but 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 that 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 = Ax, 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 vector u and the vector space X of vector 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 noninvariant results. In this article the problem of inconsistent, noninvariant solutions x(s) to u = Ax in robotics is briefly reviewed and a general theory for computing consistent, gauge-invariant solutions to nonhomogeneous systems of the form u = Ax 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.