Dynamic properties of velocity fields encoding contouring tasks for feedback control design

In this paper, the author analyzes the dynamic properties of Velocity Fields when they are used to code contouring tasks for feedback control design. These properties determine the expected motion of a system immersed in the field, forced to follow its flow lines, and introduce a formal approach to establish the connection between the motion in the Cartesian space and its analytic description in the form of a sliding mode. The analysis is intended to provide control engineers of supplementary information about the intuitive description of the desired motion described by the Velocity Fields and the dynamic induced in the system when they are used as a reference to improve the control design. Starting with a description of the Vector Fields designed for contouring tasks, the author discusses about their structure emulating the variable structure systems to force the closed-loop dynamics, with respect to the tracking velocity error, to approach to and move on the contouring surface where the flow lines of the field define a sliding condition of the contouring and tracking errors where the largest invariant set is the desired path on the contouring surface. Finally, he illustrates the complementary use of these dynamic properties for control purposes analyzing the stability of a general mechanical system in a closed loop configuration with a simple control law proportional to the tracking velocity error.