Singularities in the Image-Based Visual Servoing of Five Points.

Image-based Visual Servoing (IBVS) hinges on the observation of geometric entities such as 3D points on a target object and their projected image on the camera to control a robot. It includes a mapping between two vector spaces: the time variation of the projected 2D points on the image plane of the camera and the instantaneous spatial velocity of the camera induced by the motion of the robot. The configuration of the camera and the points is said to be singular if the matrix that maps the two vector spaces, the so-called interaction matrix, loses rank, leading to problems in control. Utilizing the results on singularities in visual servoing of three and four 3D points, the goal of this article is to determine the configuration of five points leading to singularities. The problem boils down to examining the vanishing conditions of all the minors of the interaction matrix. It turns out that it is not necessary to consider all the minors. Thus, the simplest set of equations is considered and tools from computational algebraic geometry are used to derive a necessary condition leading to singularities in IBVS of five points.