Visual servoing of robot manipulators part I: Projective kinematics

Visual servoing of robot manipulators is a key technique where the appearan ce of an object in the image plane is used to control the velocity of the e nd-effector such that the desired position is reached in the scene. The vas t majority of visual servoing methods proposed so far uses calibrated robot s in conjunction with calibrated cameras. It has been shown that the behavi or of visual control loops does not degrade too much in the presence of cal ibration errors. Nevertheless, camera and robot calibration are complex and time-consuming processes requiring special-purpose mechanical devices, suc h as theodolites and calibration jigs. In this paper; we suggest formulating a visual servoing control loop in a n onmetric space, which in our case amounts to the projective space in which a triangulation of the scene using an uncalibrated stereo rig is expressed. The major consequence of controlling the robot in nonmetric space rather t han in Euclidean space is that both the robot's direct kinematic map and th e robot's Jacobian matrix must be defined in this space as well. The elementary joint-space motions that can be performed by a robot manipul ator are pure rotations and pure translations. Traditionally, these motions are represented as Euclidean transformations. Since these motions are obse rved with an uncalibrated stereo rig, it will be convenient to represent th em as projective transformations (homographies) rather than Euclidean trans formations. Indeed, it will be shown that rotations and translations can be parameterized as special cases of homographies, which will be called proje ctive rotations and projective translations. The algebraic properties of th is nonmetric representation of elementary motions will be thoroughly invest igated allowing us to characterize the direct kinematic map and the Jacobia n matrix of a manipulator: Therefore, we introduce the concepts of projecti ve kinematics and a projective Jacobian matrix. Unlike the classical robot Jacobian matrix of a manipulator that relates the robot joint-velocities to the kinematic screw associated with the end-effector we establish a direct relationship between joint-velocities and image-plane velocities. The fatt er are velocities associated with image points arising from the 3-D to 2-D projection of end-effector points. Finally, we provide a practical method to estimate the projective kinematic model and we describe some preliminary simulated experiments that use this nonmetric model to perform stereo-based servoing. Nevertheless in-depth an alysis of projective control will be the topic of a forth coming paper.