Designing method of Passive Velocity Fields for control purposes based on Fuzzy interpolation

A Passive Velocity-Field is a vector field defining a passive motion task in terms of velocity, i.e. a kinematic description of a desired trajectory to be followed by a mechanical system, and the attracting direction to reach it. This kind of fields was defined for control purposes to attain a high degree of coordination and synchrony between their different degrees of freedom of full actuated mechanical systems when, a close pursuing of a desired trajectory, is more important than a perfect temporal tracking. The form of the field is highly dependent on the task and it is claimed not to be trivial to obtain, that is why, in this paper we face the problem of designing Passive Velocity-Fields, and we introduce a general method based on concepts of differential geometry and of the Fuzzy Logic and Fuzzy Subsets Theory. This method combines into a unique and continuous field the guiding and attracting actions of a desired trajectory, defined, respectively, from its tangent and its normal vectors, combined thanks to the use of a Fuzzy approach that provides a continuous and smooth transition between these actions keeping the passivity of the Field by the witty definition of the membership functions.