Velocity field construction for contour following tasks represented in NURBS form

Recently, a number of studies on encoding desired contour by velocity field have been reported. Since the constructed velocity field is a function of position only, the design of motion controllers for contour following can place emphasis on the reduction of contour errors rather than the reduction of tracking errors. However, very often a distance metric needs to be computed when constructing velocity fields. Though finding the corresponding distance metric for line or circle contour following tasks is relatively easy, it is not so easy to find for free form contours such as Non-Uniform Rational B-Splines (NURBS). In order to cope with this problem, this note exploits the idea of direction field theory to construct the velocity fields for free form contour following tasks. Detailed proofs of the theorems related to the proposed approach are given. Simulation results demonstrate the effectiveness of the proposed approach.