On the structure of minimum effort solutions with application to kinematic redundancy resolution

There exist in robotics, as in many other disciplines, problems described by an underdetermined set of constraints, possessing an infinite number of solutions. The problem of robot manipulator redundancy resolution is just such a situation, requiring that a particular solution be chosen according to some type of optimization criterion. One possibility employs a type of optimization which minimizes the maximum magnitude of the solution vector. This is the minimum infinity norm solution, also known as the “minimum effort solution.” This paper explores the details of least infinity norm optimization, using kinematic redundancy resolution as a test case to explore the details of infinity-norm optimization. We introduce for the first time a closed-form expression for minimum effort solutions, illustrating the heretofore unknown properties of nonuniqueness and discontinuity in time-varying situations, and postulating a possible remedy for the discontinuity problem. Additionally, to reinforce the mathematics, simulations of four-link robots are included, as well as an extended discussion of minimum-effort solutions from a geometric point of view.