The online solution of the hand-eye problem

The hand-eye problem consists in determining the relative pose between two coordinate frames fixed to the same rigid body from measurements of the poses attained by these two frames, as the body moves. In robotics this problem arises when two frames are attached to the end-effector (EE), one of these at the gripper, the other to a sensor such as a camera or a laser range-finder. Various procedures have been proposed to solve this problem when perfect pose measurements are available at a pair of EE poses, the treatment of noisy measurements being a current research topic. Solutions proposed for the case of perfect measurements require an iterative procedure based on the singular-value decomposition, which itself relies on iterative procedures. The treatment of noisy measurements has led to offline least-square solutions. It is shown in this paper that, based on an invariant formulation of the problem at hand, a solution is possible that relies on recursive linear least squares. Thus, the procedure lends itself to an online implementation, as demonstrated here with experimental results. A major difference between the proposed procedure and those reported in the literature is that the latter are iterative; ours is recursive.