A new analytical method to resolve underspecified systems of algebraic equations is presented. The method is referred to as the full space parameterization (FSP) method and utilizes easily-calculated projected solution vectors to generate the entire space of solutions of the underspecified system. Analytic parameterizations for both the space of solutions and the null space of the system reduce the determination of a task-requirement-based single solution to a m-n dimensional problem, where m-n is the degree of underspecification, or degree of redundancy, of the system. An analytical solution is presented to directly calculate the least-norm solution from the parameterized space and the results are compared to solutions of the standard pseudo-inverse algorithm which embodies the (least-norm) Moore-Penrose generalized inverse. Application of the new solution method to a variety of systems and task requirements are discussed, and sample results using four-link planar manipulators with one or two degrees of redundancy and a seven degree-of-freedom manipulator with one or four degrees of redundancy are presented to illustrate the efficiency of the new FSP method and algorithm.
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