Visual Servoing of Robot Manipulators Part I: Projective Kinematics

Visual servoing of robot manipulators is a key technique where video images, i.e. the visual appearance of an object is used to control the end-e ector. Most such methods use robots and cameras that have been calibrated. In this report we suggest a non-metric formulation of visual servoing that works with an uncalibrated stereo rig, i.e. in the projective space associated with it. In consequence, the robot kinematics and the robot Jacobian have to de ned projectively, as well. Elementary motions of robot joints are pure rotations and pure translations. Rather than representing them by Euclidean transformations, the uncalibrated stereo rig suggests to represent them by corresponding projective transformations (homographies). They are introduced as projective rotations and projective translations, their algebraic properties and special parameterizations are thoroughly investigated. This gives raise to the de nition of a projectiv kinematic map and a projective Jacobian matrix. Unlike the classical robot Jacobian, which relates joint-velocities to end-e ector velocities, we establish a direct relationship between joint-velocities and image-velocities, i.e. the velocities of those images points corresponding to end-e ector features. Finally, we provide a practical method to establish the projective kinematics and describe preliminary simulated experiments of stereo-based visual servoing using the non-metric model. Nevertheless, in depth analysis of projective control will be the topic of a forthcoming paper. Key-words: projective geometry, uncalibrated vision, kinematics, stereo vision, articulated motion, visual servoing