The authors describe a novel technique for computing position and orientation of a camera relative to the last joint of a robot manipulator in an eye-on-hand configuration. It takes only about 100+64N arithmetic operations to compute the hand/eye relationship after the robot finishes the movement, and incurs only additional 64 arithmetic operations for each additional station. The robot makes a series of automatically planned movements with a camera rigidly mounted at the gripper. At the end of each move, it takes a total of 90 ms to grab an image, extract image feature coordinates, and perform camera extrinsic calibration. After the robot finishes all the movements, it takes only a few milliseconds to do the calibration. A series of generic geometric properties or lemmas are presented, leading to the derivation of the final algorithms, which are aimed at simplicity, efficiency, and accuracy while giving ample geometric and algebraic insights. Critical factors influencing the accuracy are analyzed, and procedures for improving accuracy are introduced. Test results of both simulation and real experiments on an IBM Cartesian robot are reported and analyzed. >