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On polynomial vector fields having a given affine variety as attractive and invariant set: application to robotics

The main goal of this paper is to compute a class of polynomial vector fields, whose associated dynamical system has a given affine variety as attractive and invariant set, a given point in such an affine variety as invariant and attractive and another given affine variety as invariant set, solving the application of this technique in the robotic area. This objective is reached by using some tools taken from algebraic geometry. Practical examples of how these vector fields can be computed are reported. Moreover, by using these techniques, two feedback control laws, respectively, for a unicycle-like mobile robot and for a car-like mobile robot, which make them move, within the workspace, approaching to a selected algebraic curve, are given.