Sphere-world

Abstract: 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 E/sup n/ 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. >

Abstract: 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 E/sup n/ 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. >

Abstract: This paper introduces an algorithm for automatically tuning analytic navigation functions for sphere worlds. The tuning parameter must satisfy a lower bound to ensure collision avoidance and convergence. Until now analytic navigation functions have been manually tuned, although existence of a lower bound had been proved. A theoretical improvement on this lower bound is provided and the method is extended to unbounded manifolds. Then the required formulas are derived and algorithm described. So the lower bound is here evaluated in terms of sphere world centers and radii. Automated tuning enables completely unattended solution of any navigation problem in unknown sphere worlds and a priori known worlds which belong to the sphere world diffeomorphism class.

Abstract: In this paper we present a polynomial navigation function (NF) for a sphere world that can be constructed almost locally, with partial knowledge of the environment. The presented navigation function is C2 and as a result the computational complexity is very low, while the construction uses local knowledge and information. Moreover, an almost locally computable diffeomorphism between convex obstacles and spheres is presented, allowing the NF scheme to be used in a workspace populated by convex obstacles. Our approach is not strictly local in the epsiv sense, i.e., the field around a point is not influenced only by an e region around the point, but rather it is local in the sense that the NF around each obstacle is influenced only by the obstacle and the adjacent obstacles. In particular, we require, in the vicinity of an obstacle, the distance between the obstacle and the adjacent obstacles. Simulations are presented to verify this approach.