The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on Star Worlds

A Euclidean Sphere World is a compact connected submanifold of Euclidean n-space whose boundary is the disjoint union of a finite number of (n — 1) dimensional Euclidean spheres. A Star World is a homeomorph of a Euclidean Sphere World, each of whose boundary components forms the boundary of a star shaped set. We construct a family of analytic diffeomorphisms from any analytic Star World to an appropriate Euclidean Sphere World model. Since our construction is expressed in closed form using elementary algebraic operations, the family is effectively computable. The need for such a family of diffeomorphisms arises in the setting of robot navigation and control. We conclude by mentioning a topological classification problem whose resolution is critical to the eventual practicability of these results

The Construction of Analytic Diffeomorphisms for Exact Robot Navigation on Star Worlds

A navigation function is a scalar valued function on a robot configuration space which encodes the task of moving to a desired destination without hitting any obstacles. Our program of research concerns the construction of navigation functions on a family of configuration spaces whose “geometric expressiveness” is rich enough for navigation amidst real world obstacles. A sphere world is a compact connected subset of En whose boundary is the finite union of disjoint (n-1)-spheres. In previous work we have constructed navigation functions for every sphere world. In this paper we embark upon the task of extending the construction of navigation function to “star worlds.” A star world is a compact connected subset of En obtained by removing from a compact star shaped set a finite number of smaller disjoint open star shaped sets.This paper introduces a family of transformations from any star world into a suitable sphere world model, and demonstrates that these transformations are actually analytic diffeomorphisms. Since the defining properties of navigation functions are invariant under diffeomorphism, this construction, in composition with the previously developed navigation function on the corresponding model sphere world, immediately induces a navigation function on the star world. For more information: Kod*La

Conformal Navigation Transformations with Application to Robot Navigation in Complex Workspaces

Navigation functions provide both path and motion planning, which can be used to ensure obstacle avoidance and convergence in the sphere world. When dealing with complex and realistic scenarios, constructing a transformation to the sphere world is essential and, at the same time, challenging. This work proposes a novel transformation termed the conformal navigation transformation to achieve collision-free navigation of a robot in a workspace populated with obstacles of arbitrary shapes. The properties of the conformal navigation transformation, including uniqueness, invariance of navigation properties, and no angular deformation, are investigated, which contribute to the solution of the robot navigation problem in complex environments. Based on navigation functions and the proposed transformation, feedback controllers are derived for the automatic guidance and motion control of kinematic and dynamic mobile robots. Moreover, an iterative method is proposed to construct the conformal navigation transformation in a multiply-connected workspace, which transforms the multiply-connected problem into multiple simply-connected problems to achieve fast convergence. In addition to the analytic guarantees, simulation studies verify the effectiveness of the proposed methodology in workspaces with non-trivial obstacles

Autonomous Mobile Robots Controlled by Navigation Functions

This paper reviews the theory of navigation functions and the attendant use of natural control techniques with emphasis upon applications to mobile autonomous robots. Results to date will be discussed in the context of a larger program of research that seeks effective parameterizations of uncertainty in robot navigation problems. Constructive solutions to particular cases of mobile robot navigation problems with complete certainty are provided as well

Reactive Navigation in Partially Known Non-Convex Environments

This paper presents a provably correct method for robot navigation in 2D environments cluttered with familiar but unexpected non-convex, star-shaped obstacles as well as completely unknown, convex obstacles. We presuppose a limited range onboard sensor, capable of recognizing, localizing and (leveraging ideas from constructive solid geometry) generating online from its catalogue of the familiar, non-convex shapes an implicit representation of each one. These representations underlie an online change of coordinates to a completely convex model planning space wherein a previously developed online construction yields a provably correct reactive controller that is pulled back to the physically sensed representation to generate the actual robot commands. We extend the construction to differential drive robots, and suggest the empirical utility of the proposed control architecture using both formal proofs and numerical simulations. For more information: Kod*la