The construction of analytic diffeomorphisms for exact robot navigation on star worlds

The authors consider the construction of navigation functions on configuration spaces whose geometric expressiveness is rich enough for navigation amidst real-world obstacles. They describe a general methodology which extends the construction of navigation functions on sphere worlds to any smoothly deformable space. According to this methodology, the problem of constructing a navigation function is reduced to the construction of a transformation mapping a given space into its model sphere world. The transformation must satisfy certain regularity conditions guaranteeing invariance of the navigation function properties. The authors demonstrate this idea by constructing navigation functions on star worlds: n-dimensional star shaped subsets of Eⁿ punctured by any finite number of smaller disjoint n-dimensional stars. This construction yields automatically a bounded torque feedback control law which is guaranteed to guide the robot to destination point from almost every initial position without hitting any obstacle