• visual servoing
• optimal control
• motion planning

This thesis focuses on the characterization of optimal (shortest) paths to a desired position for a robot with unicycle kinematics and an on-board camera with limited Field-Of-View (FOV), which must keep a given feature in sight. In particular, I provide a complete optimal synthesis for the problem, i.e., a language of optimal control words, and a global partition of the motion plane induced by shortest paths, such that a word in the optimal language is univocally associated to a region and completely describes the shortest path from any starting point in that region to the goal point. Moreover, I provide a generalization to the case of arbitrary FOVs, including the case that the direction of motion is not an axis of symmetry for the FOV, and even that it is not contained in the FOV.

Finally, based on the shortest path synthesis available, feedback control laws are defined for any point on the motion plane exploiting geometric properties of the synthesis itself. Moreover, by using a slightly generalized stability analysis setting, which is that of stability on a manifold, a proof of stability is given for the controlled system. At the end, simulation results are reported to demonstrate the effectiveness of the proposed technique.