Properties of minimum infinity-norm optimization applied to kinematically redundant robots

There are numerous situations in robotics where it becomes desirable to minimize the maximum magnitude of a solution to an underdetermined set of linear equations. For example, there have been several approaches to finding the joint velocities of kinematically redundant robots using this philosophy. Unfortunately, the solution of this optimization problem, known as the “minimum infinity-norm solution”, cannot be expressed in a closed form in general thus requiring the use of an algorithm to iteratively refine an initial guess before reaching the desired solution. In order to increase our understanding and reduce the complexity of infinity-norm algorithms, we first formulate a new “infinity inverse”, and then use the new inverse to explore critical issues such as uniqueness and continuity of least infinity-norm solutions. The new inverse is compared with the well-known pseudoinverse, or minimum two-norm solution. We discuss when and why one particular norm might produce a better solution than the other, reinforcing the discussion with an interesting example of a kinematically redundant manipulator