Navigation Functions for everywhere partially sufficiently curved worlds

Abstract

Abstract-We extend Navigation Functions (NF) to worlds of more general geometry and topology. This is achieved without the need for diffeomorphisms, by direct definition in the geometrically complicated configuration space. Every obstacle boundary point should be partially sufficiently curved. This requires that at least one principal normal curvature be sufficient. A normal curvature is termed sufficient when the tangent sphere with diameter the associated curvature radius is a subset of the obstacle. Examples include ellipses with bounded eccentricity, tori, cylinders, one-sheet hyperboloids and others. Our proof establishes the existence of appropriate tuning for this purpose. Direct application to geometrically complicated cases is illustrated through nontrivial simulations.

Navigation functions for focally admissible surfaces

“…To avoid the need for diffeomorphisms, the authors pro posed an extension in [14] which introduced a second-order geometric condition characterizing the surfaces for which a Koditschek-Rimon NF (KRNF) exists.…”

Navigation functions learning from experiments: Application to anthropomorphic grasping

“…Hence, if the experimental speeds do not become zero, this is mathematically guaranteed to yield a NF for the configuration space subset that has been experimentally explored. The extension in [21] justifies the application of KRNFs to more complicated environments.…”

“…When F is an everywhere partially sufficiently curved world, as defined in [21], then there exists ak min > 0, such that ϕ in Eq. (2) is a NF for all k >k min [21]. Sphere worlds are a special sub-category.…”

“…These map the actual C-space obstacles to spheres. We have recently extended NFs to more general geometries in [21], without the need of diffeomorphisms. This enables the use of more complicated obstacles, like tori, ellipsoids and partially non-convex surfaces, as long as a curvature condition is satisfied.…”

Collaborative Multi-Robot Transportation in Obstacle-Cluttered Environments via Implicit Communication

“…Apparently, the overall formation has to be aligned with the x-axis in order to transverse the obstacles. In this respect, we constructed a navigation function in a 3D workspace (i.e., x, y, and ψ), by adopting a virtual toroidal obstacle (see Filippidis and Kyriakopoulos, 2012 for the safety and convergence properties) to model the aforementioned relation of position x, y with the orientation ψ, as depicted in Figure 10. Moreover, the control gains were selected as follows: k = 1.9, k NF = 0.8, K L = 3I 3×3 , = 3I 3×3 , K F = 3I 3×3 .…”