Relationship between the Pseudoinverse formulation and the Compact formulation for obtaining the general solutions of redundant robotic systems

The general solutions of redundant robotic systems for Force Distribution and Redundant Inverse Kinematics problems have been formulated using the traditional Pseudoinverse formulation and the Compact formulation. The Compact formulation is derived by applying Gaussian elimination with partial pivoting to transform the underspecified matrix into a row-reduced echelon form; then, the general solution is still formulated as one particular solution together with a homogeneous solution. The major difference is that, unlike the Pseudoinverse formulation, the homogeneous solution of the Compact formulation is merely a function of the free variables. As such, the optimization problem size is greatly reduced so that the computational burden (especially for constrained optimization problems with inequalities) is also greatly reduced. In this paper, the relationship between these two formulations is investigated. It is shown that the Pseudoinverse formulation can be converted into the Compact formulation with the free variables chosen as a function of the arbitrary vector of the Pseudoinverse formulation.