Enhanced code for the full space parameterization approach to solving underspecified systems of algebraic equations: Version 1.0

摘要

This paper describes an enhanced version of the code for the Full Space Parameterization (FSP) method that has recently been presented for determining optimized (and possibly constrained) solutions, x, to underspecified system's of algebraic equations b = Ax. The enhanced code uses the conditions necessary for linear independence of the m (minus) n + 1 vectors forming the solution as a basis for an efficient search pattern to quickly find the full set of solution vectors. A discussion is made of the complications which may be present due to the particular combination of the matrix A and the vector b. The first part of the code implements the various methods needed to handle these particular cases before the solution vectors are calculated so that computation time may be decreased. The second portion of the code implements methods which can be used to calculate the necessary solution vectors. The respective expressions of the full solution space, S, for the cases of the matrix A being full rank and rank deficient are given. Finally, examples of the resolution of particular cases are provided, and a sample application to the joint motion of a mobile manipulator for a given end-effector trajectory is presented.