Visual servoing of robot manipulators part I: Projective kinematics

Visual servoing of robot manipulators is a key technique where the appearance of an object in the image plane is used to control the velocity of the end-effector such that the desired position is reached in the scene. The vast majority of visual servoing methods proposed so far uses calibrated robots in conjunction with calibrated cameras. It has been shown that the behavior of visual control loops does not degrade too much in the presence of calibration errors. Nevertheless, camera and robot calibration are complex and time-consuming processes requiring special-purpose mechanical devices, such as theodolites and calibration jigs. In this paper; we suggest formulating a visual servoing control loop in a nonmetric 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 than in Euclidean space is that both the robot's direct kinematic map and the robot's Jacobian matrix must be defined in this space as well. The elementary joint-space motions that can be performed by a robot manipulator are pure rotations and pure translations. Traditionally, these motions are represented as Euclidean transformations. Since these motions are observed with an uncalibrated stereo rig, it will be convenient to represent them as projective transformations (homographies) rather than Euclidean transformations. Indeed, it will be shown that rotations and translations can be parameterized as special cases of homographies, which will be called projective rotations and projective translations. The algebraic properties of this nonmetric representation of elementary motions will be thoroughly investigated allowing us to characterize the direct kinematic map and the Jacobian matrix of a manipulator: Therefore, we introduce the concepts of projective 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 fatter 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 analysis of projective control will be the topic of a forth coming paper.