Determination of All Stable and Unstable Equilibria for Image-Point-Based Visual Servoing

Local minima are a well-known drawback of image-based visual servoing systems. Up to now, there were no formal guarantees on their number, or even their existence, according to the considered configuration. In this work, a formal approach is presented for the exhaustive computation of all minima and unstable equilibria for a class of six well-known image-based visual servoing controllers. This approach relies on a new polynomial formulation of the equilibrium condition that avoids using the camera pose. By using modern computational algebraic geometry methods and an ad hoc symmetry breaking strategy, the formal resolution of this new equilibrium condition is rendered computationally feasible. The proposed methodology is applied to compute the equilibria of several classical visual servoing tasks, with planar and nonplanar configurations of four and five points. The effects of local minima and saddle points on the dynamics of the system are finally illustrated through intensive simulation results, as well as the effects of image noise and uncertainties on depths.