Robot Navigation Functions on Manifolds with Boundary

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are constructed on an arbitrary sphere world—a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n − l)-spheres. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider. For more information: Kod*La

Exact Robot Navigation Using Artificial Potential Functions

We present a new methodology for exact robot motion planning and control that unifies the purely kinematic path planning problem with the lower level feedback controller design. Complete information about the freespace and goal is encoded in the form of a special artificial potential function - a navigation function - that connects the kinematic planning problem with the dynamic execution problem in a provably correct fashion. The navigation function automatically gives rise to a bounded-torque feedback controller for the robot\u27s actuators that guarantees collision-free motion and convergence to the destination from almost all initial free configurations. Since navigation functions exist for any robot and obstacle course, our methodology is completely general in principle. However, this paper is mainly concerned with certain constructive techniques for a particular class of motion planning problems. Specifically, we present a formula for navigation functions that guide a point-mass robot in a generalized sphere world. The simplest member of this family is a space obtained by puncturing a disc by an arbitrary number of smaller disjoint discs representing obstacles. The other spaces are obtained from this model by a suitable coordinate transformation that we show how to build. Our constructions exploit these coordinate transformations to adapt a navigation function on the model space to its more geometrically complicated (but topologically equivalent) instances. The formula that we present admits sphere-worlds of arbitrary dimension and is directly applicable to configuration spaces whose forbidden regions can be modeled by such generalized discs. We have implemented these navigation functions on planar scenarios, and simulation results are provided throughout the paper

Exact robot navigation using cost functions: the case of distinct spherical boundaries in E\u3csup\u3en\u3c/sup\u3e

The utility of artificial potential functions is explored as a means of translating automatically a robot task description into a feedback control law to drive the robot actuators. A class of functions is sought which will guide a point robot amid any finite number of spherically bounded obstacles in Euclidean n-space toward an arbitrary destination point. By introducing a set of additional constraints, the subclass of navigation functions is defined. This class is dynamically sound in the sense that the actual mechanical system will inherit the essential aspects of the qualitative behavior of the gradient lines of the cost function. An existence proof is given by constructing a one parameter family of such functions; the parameter is used to guarantee the absence of local minima

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

Exact robot navigation in geometrically complicated but topologically simple spaces

A navigation function is an artificial potential energy function on a robot configuration space (C-space) which encodes the task of moving to an arbitrary destination without hitting any obstacle. In particular, such a function possesses no spurious local minima. In this paper we construct navigation functions on forests of stars: geometrically complicated C-spaces that are topologically indistinguishable from a simple disc punctured by disjoint smaller discs, representing model obstacles. For reasons of mathematical tractability we approximate each C-space obstacle by a Boolean combination of linear and quadratic polynomial inequalities (with sharp corners allowed), and use a calculus of implicit representations to effectively represent such obstacles. We provide evidence of the effectiveness of this technology of implicit representations in the form of several simulation studies illustrated at the end of the paper

A General Stance Stability Test Based on Stratified Morse Theory With Application to Quasi-Static Locomotion Planning

This paper considers the stability of an object supported by several frictionless contacts in a potential field such as gravity. The bodies supporting the object induce a partition of the object's configuration space into strata corresponding to different contact arrangements. Stance stability becomes a geometric problem of determining whether the object's configuration is a local minimum of its potential energy function on the stratified configuration space. We use Stratified Morse Theory to develop a generic stance stability test that has the following characteristics. For a small number of contacts---less than three in 2-D and less than six in 3-D---stance stability depends both on surface normals and surface curvature at the contacts. Moreover, lower curvature at the contacts leads to better stability. For a larger number of contacts, stance stability depends only on surface normals at the contacts. The stance stability test is applied to quasi-static locomotion planning in two dimensions. The region of stable center-of-mass positions associated with a $k$-contact stance is characterized. Then, a quasi-static locomotion scheme for a three-legged robot over a piecewise linear terrain is described. Finally, friction is shown to provide robustness and enhanced stability for the frictionless locomotion plan. A full maneuver simulation illustrates the locomotion scheme