Stability Analysis of Plane-to-Plane Positioning by Proximity-Based Control

In this letter, we discuss the stability analysis of Plane-to-Plane positioning task when the task is designed in proximity sensor space. We utilize a multi-sensor arrangement of proximity sensors that forms a proximity array to obtain the necessary information in sensor space. For the task considered, we provide closed-form equations for the closed-loop system by obtaining the analytical form of pseudo-inverse for the interaction matrix involved. This further enables us to suggest a new control law producing a decoupled exponential decrease of the sensor errors in perfect conditions, while being more robust to estimation errors in the surface normal. By applying Gershgorin's theorem to the closed-form matrices, we are able to provide conditions for stability with respect to errors in extrinsic parameters and surface normal. Simulation results are provided to discuss the robustness of the task with respect to these modeling parameters.